POLYGONAL LINES INSCRIBED AND CIRCUMSCRIBED WITH A CONVEX ARC

BY
TIBERIU POPOVICIU
Romania

Section 1. Preliminary Questions

1.

G. Tzitzéica was not only a prestigious scholar due to his remarkable research in geometry, but also an eminent professor who contributed significantly to the education of many generations of mathematicians in Romania. An active contributor to the journal "Gazeta Matematica," to which many of us owe our introduction to mathematics, G. Tzitzéica wrote numerous articles and notes on "elementary" topics, but always interesting and insightful.
2.

In a short note, G. Tzitzéica proposes [5] an Elementary Proof of the Inequality

\sin x>\frac{2x}{\pi}\text{ where }0<x<\frac{\pi}{2},

(1)

based on the remark that

\operatorname{arc}MAM^{\prime}>\operatorname{arc}MOM^{\prime}

(2)

Or $O$ is the origin of arcs on the unit circle $C,M,M{}^{\prime}$ are the extremities of

bows $x,-x$ respectively and the are $MAMM^{\prime}$ East $1a$ length of the semicircle deorit on $MM^{\prime}$ as diameter (see fig. 1).

According to DS Mitrinović, inequality (1) is due to Jordan [2].

Lia's proof of inequality (2), which is not given by G. Tzitzéica, suggests a more general property relating to two plane convex arcs, one enveloping the other. Indeed, arcs of circles $MAMM^{\prime}$ And $MODM^{\prime}$ are indeed convex, and the first encloses the second. In what follows, we will highlight the property that the length of the enclosing arc is greater than that of the enclosed arc.
3. We will begin by studying some relationships between the elements of a triangle, relationships that we will use later.

Let's consider a triangle $ABO$ whose vertices, in the plane referred to two rectangular axes $Oxy$ , are the points $B\left(x_{1},f\left(x_{1}\right)\right),A\left(x_{2}\right.$ ,

$\left.f\left(w_{2}\right)\right),C\left(w_{3},f\left(w_{3}\right)\right)$ We will assume that the x-coordinates are numbered in order $x_{1}<x_{2}<x_{3}$ The ordinates are the respective values of a function $f$ defined on the points $x_{1},x_{2}$ , $w_{3}$ In what follows, we will only consider triangles in this situation, but it is easy to see that the results we obtain can be transposed to any triangle in the plane, by choosing the coordinate axes appropriately.
Let us denote by $ABC$ the lengths of the opposite sides, respectively at the vertices $ABC$ and by $d$ the length of the segment $AD$ Or $D$ is the point where the ordinate of the point $HAS$ cut the side $BC$ ( $D$ is between $B$ And $O$ ).

The length of the straight line segment with endpoints ( $x^{\prime},f\left(x^{\prime}\right)$ ), ( $x^{\prime\prime},f\left(x^{\prime\prime}\right)$ ) is equal to $\sqrt{\left(x^{\prime}-x^{\prime\prime}\right)^{2}+\left(f\left(x^{\prime}\right)-f\left(x^{\prime\prime}\right)\righ t)^{2}}=\left|x^{\prime}-x^{\prime\prime}\right|\sqrt{1+\left\{\left[x^{\prime},x^{\prime\prime};f\right]\right\}^{2}}$ where we have designated by

\left[x^{\prime}w^{\prime\prime};f\right]=\frac{f\left(w^{\prime\prime}\right)-f\left(w^{\prime}\right)}{w^{\prime\prime}-w^{\prime}}

(3)

the divided difference (of the first order) of the function $f$ on the points (or nodes) $\mathfrak{x}^{\prime},\mathfrak{x}^{\prime\prime}$ .

Let's return to the triangle. $ABC$ and let us assume, to simplify the notation, $\alpha=\left[x_{1},x_{2};f\right],\beta=\left[x_{2},x_{3};f\right],\gamma=\left[x_{1},x_{3};f\right]$ We then have

	$\displaystyle a=\left(x_{3}-x_{1}\right)\sqrt{1+\gamma^{2}},b=\left(x_{3}-x_{2}\right)\sqrt{1+\beta^{2}}$
	$\displaystyle c=\left(x_{2}-x_{1}\right)\sqrt{1+\alpha^{2}}$		(4)

Using these formulas, we can establish some inequalities that we will use later.
4. Triangular inequalities $bc<a\leqq b+c$ are equivalent to inequalities

(bc)^2 < a^2 \leqq(b+c)^2

(5)

and formulas
(6) result $\quad(b+c)^{2}-a^{2}=2\left(w_{2}-x_{1}\right)\left(w_{3}-x_{2}\right)\left(\sqrt{1+\alpha^{2}}\sqrt{1+\beta^{2}}-1-\alpha\beta\right)$
(7) $\quad a^{2}-(b-c)^{2}=2\left(x_{2}-x_{1}\right)\left(x_{3}-x_{2}\right)\left(\sqrt{1+\alpha^{2}}\sqrt{1+\beta^{2}}+1+\alpha\beta\right)$ .

They are obtained from (4) and the formula for the mean of the divided differences.

\gamma=\frac{\left(x_{2}-x_{1}\right)\alpha+\left(x_{3}-x_{2}\right)\beta}{x_{3}-x_{1}}.

(8)

If we notice that we have inequality $\left(1+\alpha^{2}\right)\left(1+\beta^{2}\right)\geqq(1++\alpha\beta)^{2}$ , where equality occurs if and only if $\alpha=\beta$ , we deduce

\sqrt{\left(1+\alpha^{2}\right)\left(1+\beta^{2}\right)}-1-\alpha\beta\geqq 0,\sqrt{\left(1+\alpha^{2}\right)\left(1+\beta^{2}\right)}+1+\alpha\beta\geqq 0

Taking into account (6) and (7), inequalities (5) follow.
Note, in passing, that we also have
$\left(1+\alpha^{2}\right)\left(1+\beta^{2}\right)\geqq(1-\alpha\beta)^{2}$ where equality occurs if, and only if, $\alpha+\beta=0$ It follows that

\sqrt{\left.1+\alpha^{2}\right)\left(1+\beta^{2}\right)}+1+\alpha\beta\geqq 2,

(9)

inequality which will be used later.
From inequalities (5) it also results

(b+c)^{2}\leqq\left(1+\frac{(b+c)^{2}-a^{2}}{a^{2}-(b-c)^{2}}\right)a^{2}

(10)

the verification of which can be left to the reader*.
Taking into account (6), (7) and (9) we deduce

		$\displaystyle\frac{(b+c)^{2}-a^{2}}{a^{2}-(b-c)^{2}}=\frac{\sqrt{\left(1+\alpha^{2}\right)\left(1+\beta^{2}\right)}-1-\alpha\beta}{\sqrt{\left(1+\alpha^{2}\right)\left(1+\beta^{2}\right)}+1+\alpha\beta}=$
		$\displaystyle=\frac{(\alpha-\beta)^{2}}{\left(\sqrt{\left(1+\alpha^{2}\right)\left(1+\beta^{2}\right)}+1+\alpha\beta\right)^{2}}\leqq\frac{(\alpha-\beta)^{2}}{4}.$

•

Inequality comes down to $b+c\leqslant\frac{a}{\sin\frac{A}{2}}$ , whose trigonometric demonstration is immediate and in turn leads back to $\cos\frac{B-C}{2}\leqq 1$ .

Note that $1+\frac{(\alpha-\beta)^{2}}{4}\leqq\left(1+\frac{|\alpha-\beta|}{2}\right)^{2}$ and then from (10) we deduce the inequality

b+c\leqq\left(1+\frac{|\alpha-\beta|}{2}\right)a.

(11)

5.

We have

a^{2}-b^{2}-e^{2}=2\left(x_{2}-x_{1}\right)\left(x_{3}-x_{2}\right)(1+\alpha\beta).

(12)

If we notice that the ordinate of the point $D$ is equal to $f\left(x_{1}\right)+\left(x_{2}-\right.\left.-x_{1}\right)$ y or $f\left(x_{3}\right)+\left(x_{2}-x_{3}\right)$ and if we take into account (8) we obtain

d^{2}=\frac{\left(x_{2}-x_{1}\right)^{2}\left(x_{3}-x_{2}\right)^{2}}{\left(x_{3}-x_{1}\right)^{2}}(\alpha-\beta)^{2}

Given (4) and (8) we deduce

d^{2}+\frac{\left(x_{2}-x_{1}\right)\left(x_{3}-x_{2}\right)}{\left(x_{3}-x_{1}\right)^{2}}a^{2}=\frac{\left(x_{2}-x_{1}\right)b^{2}+\left(x_{3}-x_{2}\right)}{x_{3}-x_{1}}

(13)

The second member is a weighted arithmetic mean of $b^{2}$ and $c^{2}$ and it follows that

d<\max(b,c)

(14)

If $|\alpha-\beta|\leqq\sqrt{2}$ We have $1+\alpha\beta\geqq 1-\frac{(\alpha-\beta)^{2}}{2}==\left(1-\frac{|\alpha-\beta|}{\sqrt{2}}\right)\left(1+\frac{|\alpha-\beta|}{\sqrt{2}}\right)\geqq 1-\frac{|\alpha-\beta|}{\sqrt{2}}\geqq 0$ done $1+\alpha\beta\geqq 0$ and then from (12) we deduce that $b^{2}+c^{2}\leqq a^{2}$ hence max $(b,c)\leqq a$ Taking into account (14), we deduct the

Lemma 1. If we have $|\alpha-\beta|\leqq\sqrt{2}$ we have inequality $d<\alpha$
We will use this property in Section 4 of this work. Mark 1.
The condition $1+\alpha\beta\geqq 0$ is equivalent to the fact that the angle $A$ of the triangle $ABC$ is not acute, therefore that $A\geqq 90^{\circ}$ .

Note 2. If we also consider the divided differences of the second order of the function $f$ defined by

\left[w^{\prime},w^{\prime\prime},w^{\prime\prime\prime};f\right]=\frac{\left[w^{\prime\prime},w^{\prime\prime\prime};f\right]-\left[w^{\prime},w^{\prime\prime};f\right]}{w^{\prime\prime\prime}-w^{\prime}}

(15)

the length $d\mathrm{du}$ segment $AD$ is equal $\left.\left(x_{2}-x_{1}\right)\left(x_{3}-x_{2}\right).\mid\left[x_{1},x_{2},x_{3};f\right]\right]$ or is given by the formula $d=\frac{2\mathcal{S}}{x_{3}-x_{1}}$ , $S$ being the area of the triangle $ABC$ .

§ 2. SOME PROPERTIES OF POLYGONAL FUNCTIONS

6.

Let us consider a bounded and closed interval $[a,b](a<b)$ on the real axis. That is $a=x_{0}<x_{1}<\ldots<x_{n-1}<x_{n}=b$ ( $n\geqq 1$ ) a division of the interval $[a,b]$ A function $P:[a,b]\rightarrow\mathbb{R}$ which is continuous and reduces over each of the partial intervals $\left[x_{i-1},x_{i}\right],i=1,2,\ldots,n$ to a polynomial of degree 1. This is called a polygonal function and its graph is a polygonal line. The points $x_{i},i=0,1,\ldots,n$ are the newds and the points $P_{i}\left(x_{i},P\left(x_{s}\right)\right)$ are the vertices of this function or polygonal line. In what follows, we will use the terms polygonal function or polygonal line, depending on the circumstances, to mean polygonal function. The polygonal function $P$ is completely determined by its values at the nodes $n_{1}$ If these values are the values taken by a function $f$ , we will denote the polygonal function by

P=P\left(x_{0},x_{1},\ldots,x_{n};f\right)

(16)

by highlighting the nodes j and j the function $f$ The polygonal function (16) is an interpolating function on the nodes $x_{i}$ and related to the function $f$ since we have

P\left(x_{i}\right)=f\left(x_{i}\right),i=0,1,\ldots,n.

The line segments $P_{t-1}P_{i}$ which connect two consecutive vertices and which are the graphs of the restrictions on the intervals $\left[x_{i-1},x_{i}\right]$ , $i=1,2,\ldots,n$ of $P$ are the sides of the function or polygonal line $P$ The length of the side $P_{i-1}P_{i}$ is well determined and is equal to

\sqrt{\left(x_{i}-x_{i-1}\right)^{2}+\left(f\left(x_{i}\right)-f\left(x_{i-1}\right)\right)^{2}}=\left(x_{i}-x_{i-1}\right)\sqrt{1+\left[x_{i-1},x_{i};f\right]^{2}}.

We will refer to $l(P)$ the length of the polygonal function $P$ , which is, by definition, equal to the sum of the lengths of its sides.

Note that the restriction of the polygonal function $P$ on a closed subinterval of $[a,b]$ is still a polygonal function. In particular, if $c$ is an interior point of $[a,b]$ the restriction $Q$ of $P$ on $[a,e]$ and the restriction $R$ of $P$ on $[c,b]$ are polygonal functions, the last node of the first coinciding with the first node of the second. We then have the property of length additivity

l(P)=l(Q)+l(R).

(17)

Conversely, if $Q:[a,b]\rightarrow\mathbb{R},R:[o,b]\rightarrow\mathbb{R}$ , Or $a<e<b$ , are polygonal functions and if $Q(c)=R(c)$ , then the function $P:[a,b]\rightarrow\mathbb{R}$ , Or

P(x)=\left\{\begin{array}[]{l}Q(x)\text{ pour }x\in[a,c]\\ R(x)\text{ pour }x\in[c,b]\end{array}\right.

is a polygonal function for which the additivity formula (17) is verified.

There are other additivity formulas. We will use those derived from (17) by repeating it a finite number of times.

Let us recall once again that we formerly called polygonal functions elementary functions of order $n$ For $n=1$ [3]. Today they are called "spline" functions. We can generalize polygonal functions in various ways. Either by taking a domain of definition more general than an interval, or by defining polygonal functions with an infinite number of nodes (see, for example, [4]). We will not use such functions here.
7. We will assume that we know the properties of convex and non-concave (first-order) functions. A function $f:E\rightarrow\mathbb{R}$ A real variable is said to be convex or non-concave if its difference divided by order 2 is positive, or non-negative if its difference divided by order 2 is positive, so if expression (15) is $>0$ , respectively $\geqq 0$ for any group of 3 distinct points $x^{\prime},x^{\prime\prime},x^{\prime\prime\prime}$ of $E$ For other definitions and properties of these functions, see my Analysis course [4].

The definition and properties of divided differences (of order 1 and 2) (3) and (15) are well known. In this work, we will use some of these properties, most often without explicitly specifying them.
8. Let us first recall the property expressed by the

Lemma 2. For the polygonal function (16), defined on the interval $[a,b]$ to be non-concave, it is necessary and sufficient that its restriction on the set $\left\{x_{i}\right\}_{i=0}^{n}$ of its nodes is not concave.

The property results immediately from the formula

	$\displaystyle P\left(x_{0},x_{1},\ldots,x_{n};f\right)=f\left(x_{0}\right)+\left[x_{0},x_{1};f\right]\left(x-x_{0}\right)+$
	$\displaystyle+\sum_{i=1}^{n-1}\frac{x_{i+1}-x_{i-1}}{2}\left[x_{i-1},x_{i},x_{i+1};f\right]\left(\left\|x-x_{i}\right\|+\infty-x_{i}\right).$		(19)

The condition of Lemma 2 is equivalent to the property according to which the sequence $\left(\left[x_{i-1},x_{i};f\right]\right)_{i=1}^{n}$ (of slopes) is non-decreasing and has the property (if $n>1$ ) according to which the sequence ( $\left.\left[x_{i-1},x_{i},x_{i+1};f\right]\right)_{i=1}^{n-1}$ successive divided differences of the second order is non-negative.

If $n=1$ , in the second member of formula (19) only the first two terms appear.

Let us designate by $\alpha_{i}$ the angle of the vector $P_{i-1}P_{i}$ with the axis $Ox$ We then have $\alpha_{i}=\operatorname{arctg}\left[x_{i-1},x_{i};f\right],-\frac{\pi}{2}<\alpha_{i}<\frac{\pi}{2},i=1,2,\ldots,n$ The difference $\alpha_{i+1}-\alpha_{i}(1\leqq i\leqq n-1)$ is the outside angle of the vertex $P_{i}$ of the polygonal line $P$ The condition of Lemma 2 is then equivalent to the property that the sequence ( $\left.\alpha_{i}\right)_{i=1}^{n}$ (of angles) is non-decreasing and has the property (if $n>1$ ) according to which the sequence ( $\left.\alpha_{i+1}-\alpha_{i}\right)_{i=1}^{n-1}$ The exterior angles are non-negative.
9. We will now demonstrate the

Lemma 3. Let $P,Q$ two polygonal functions defined on the same interval $[a,b](a<b)$ and which meet the following conditions:

1.

$P(a)=Q(a),P(b)=Q(b)$ .
2.

$P,Q$ are non-concave.
3.

We have

P\neq Q\text{ et }\forall_{x}Q(x)\leqq P(x).

(20)

So, between the lengths of $P$ And $Q$ we have inequality

l(P)<l(Q).

(21)

If the conditions $1,2,3$ of lemma 3 are verified and if $x^{\prime},x^{\prime\prime}$ , Or $\omega^{\prime}<x^{\prime\prime}$ are two consecutive nodes of the function $Q$ , of $P\left(x^{\prime}\right)=Q\left(x^{\prime}\right)$ , $P\left(p^{\prime\prime}\right)=Q\left(x^{\prime\prime}\right)$ It necessarily follows that $P(x)=Q(x)$ over the entire interval $\left[x^{\prime},x^{\prime\prime}\right]$ From conditions 1, 2, and 3, it follows that at least one of the nodes $x^{\prime}$ of $Q$ we have $Q\left(x^{\prime}\right)<P\left(x^{\prime}\right)$ That being said, it can be noted that from lemma 3 it also follows that

Lemma 3'. If $P,Q$ are two polygonal functions defined on the same interval $[a,b](a<b)$ and which satisfy conditions 1, 2 of Lemma 3 as well as the condition
$3^{\prime}$ , we have
(20')

\forall_{x}Q(x)\leqq P(x).

So, we have inequality

l(P)\leqq l(Q)

(

\prime

)

In the proof of Lemma 3 we will also use the lemma $3^{\prime}$ We still need to prove Lemma 3.
Let $n+1$ the number of nodes of $P$ And $m+1$ that of $Q.n$ And $m$ are any natural numbers and we will proceed by complete induction.

First, let us note that, by virtue of the non-concaveness of $P$ , we can exclude the case $m=1$ Indeed, if $m=1$ Given that conditions 1 and 2 are met, condition (20') can only occur if $P=Q$ and Alon's $l(P)==l(Q)$ condition 3 not being met.

We will demonstrate in two steps.

1.

First step. We will demonstrate the property for $n=1$ and for $m>1$ arbitrary. The property is true for $m=2$ In this case, then, $y_{1}$ the nowd different from $a$ and $b$ of $Q$ We then have $Q\left(y_{1}\right)<<P\left(y_{1}\right.$ , and the triangle $P_{0}Q_{1}P_{2}$ is non-degenerate. According to the arguments made above, from the triangle inequality it follows that $l(P)<lQ(Q)$ .

So now $k$ a natural number $>1$ and suppose that the property is true for all $m$ such as $1\leqq m\leqq k$ Let's demonstrate that it will also be true for $m=k+1$ So be it. $m=k+1(k>1)$ and either $y_{i}$ a knot of $Q$ such as one has $Q\left(y_{i}\right)<P\left(y_{i}\right)^{\star}$ Let us designate by $Q^{\prime}$ the restriction of $Q$ on $\left[a,y_{i}\right]$ and by $Q^{\prime\prime}$ its restriction on $\left[y_{i},b\right]$ We also refer to $P^{\prime}$ the polygonal function $P\left(a,y_{i};Q\right)$ and by $P^{\prime\prime}$ the polygonal function $P\left(y_{i},b;Q\right)$ So, according to the result relating to the case $m=2$ we have

l(P)<l\left(P^{\prime}\right)+l\left(P^{\prime\prime}\right).

(22)

Let us now note that $P^{\prime}$ And $Q^{\prime}$ are polygonal functions having respectively 2 and a number $<m$ of knots. The same applies.

⁰⁰footnotetext: • In our case, moreover, this inequality is verified for every node of

Q

different rent from

a

and

b

for polygonal functions $P^{\prime\prime}$ And $Q^{\prime\prime}$ According to the lemma $3^{\prime}$ , which by hypothesis applies in this case, we have

l\left(P^{\prime}\right)\leqq l\left(Q^{\prime}\right),l\left(P^{\prime\prime}\right)\leqq l\left(Q^{\prime\prime}\right).

(23)

But, according to the additivity of the lengths of polygonal lines, we have $l\left(Q^{\prime}\right)++l\left(Q^{\prime\prime}\right)=l(Q)$ Given (21) and (22), we deduce that $l(P)<l(Q)$ 1. What needed to be demonstrated.
2. Second step. Let's now demonstrate that for all $n$ given and whatever $m$ The property is true. For $n=1$ The property has been demonstrated above. Let's now $\%$ a natural number $>1$ and suppose that the property is true for all $n$ such as $1\leqq n\leqq k$ It suffices to demonstrate that the property is also true for $n=k+1$ Let us therefore suppose that $P$ ait $k+2$ knots and either $x_{1}$ its second knot (we have $a<a_{1}<b$ We need to examine two cases:

Case 1. $Q\left(w_{1}\right)=P\left(w_{1}\right)$ So then $P^{\prime},P^{\prime\prime}$ And $Q^{\prime},Q^{\prime\prime}$ the restric-.

tions of $P$ And $Q$ on $\left[a,x_{1}\right],\left[x_{1},b\right]$ respectively. So $P^{\prime},Q^{\prime}$ on the one hand and $P^{\prime\prime},Q^{\prime\prime}$ on the other hand, they satisfy the conditions of the lemma $3^{\prime}$ and moreover $P^{\prime}$ And $P^{\prime\prime}$ are polygonal functions of a number $<n$ of knots. With these notations, inequalities (22) are still, by hypothesis, verified. It follows that $l(P)\leqq l(Q)$ But here equality is only possible if we have equality in both relations (23), which requires $P^{\prime}=Q^{\prime},P^{\prime\prime}=Q^{\prime\prime}$ and so $P=Q$ It follows that $l(P)<l(Q)$ and lemma 3 is further proven.

Case 2. $Q\left(x_{1}\right)<P\left(x_{1}\right)$ In this case, the non-concavity of the functions $P,Q$ shows us that the function $Q$ and the linear function $y==\frac{x-a}{x_{1}-a}P\left(x_{1}\right)+\frac{x-x_{1}}{a-x_{1}}P(a)$ coincide on the interval $[a,b]$ , we are outside the point $x_{1}$ on a single point $\alpha$ which is to the right of $x_{1}$ and to the left of $b$ (see fig, 3).

We now refer to $P^{\prime},P^{\prime\prime}$ the restrictions of $P$ on $\left[a,x_{1}\right],\left[x_{1},b\right]$ respectively and by $Q^{\prime},Q^{\prime\prime}$ the restrictions of $Q$ on $[a,\alpha]$ , $[\alpha,b]$ respectively. Finally, let us designate by $P^{*}$ the polygonal function defined on [ $a,a$ ] by

P^{*}(x)=\left\{\begin{array}[]{l}P^{\prime}(x)\text{ pour }x\in\left[a,x_{1}\right]\\ y(x)\text{ pour }x\in\left[x_{1},\alpha\right],\end{array}\right.

by $Q^{*}$ the polygonal function defined on $\left[x_{1},b\right]$ by

Q^{*}(x)=\left\{\begin{array}[]{l}y(x)\text{ pour }x\in\left[x_{1},\alpha\right]\\ Q^{\prime\prime}\cdot(x)\text{ pour }x\in[\alpha,b]\end{array}\right.

and finally by $y$ the polygonal line restriction of the function $y$ on $\left[w_{1},\alpha\right]$ Note that $P^{*}$ is a polygonal function with 2 nodes (it's a straight line) and $P^{\prime\prime}$ a polygonal function to $<\eta$ knots. We therefore have, by hypothesis,

l\left(P^{*}\right)<l\left(Q^{\prime}\right),l\left(P^{\prime\prime}\right)\leqq l\left(Q^{*}\right).

(24)

But the additivity of length gives us $l(P)=l\left(P^{\prime}\right)+l\left(P^{\prime\prime}\right),l(Q)==l\left(Q^{\prime}\right)+l\left(Q^{\prime\prime}\right)$ And $l\left(P^{*}\right)=l\left(P^{\prime}\right)+l(y),l\left(Q^{*}\right)=l(y)+l\left(Q^{\prime\prime}\right)$ Taking into account (23) it follows that $l(P)<l(Q)$ In this way, lemma 3 is further demonstrated.

I must note that a special case of Lemma 3 has been demonstrated by the method indicated here by E. Hille in his interesting book on analysis [1].

§ 3. ON THE LENGTH OF INSCRIBED OR CIRCUMCROACHED POLYGONAL LINES $\mathring{\mathrm{A}}$ A CONVEX SILVER

10.

Either $f$ a convex function defined on the bounded and closed interval $[a,b]$ Or $a<b$ We know that $f$ is then continuous and has a left-hand derivative and a right-hand derivative at every point of the interior $]a,b[$ of the interval $[a,b]^{*}$ .

We will initially consider only convex functions $f:[a,b]\rightarrow\mathbb{R}$ which satisfy the following two properties:

1.

$f$ is continuous (on $[a,b]$ ),
2.

$f$ admits a (finite) left derivative, $f_{g}^{\prime}(x)$ Above all $\left.\left.x\in\right]a,b\right]$ and a right-hand derivative $f_{d}^{\prime}(x)$ (finished) on everything $w\in[a,b[$ .

To simplify, we will say that such a function is a function ( $O$ The monotonicity properties of the functions $f_{\rho}^{\prime}f_{a}^{\prime}$ are well known and there is no need to repeat them. We will use them later. Let us only note here that if $f$ is which function ( $O$ ) on the interval $[a,b]$ , its restriction to an arbitrary closed subinterval (of non-aulle length) of $[a,b]$ is also a function (C). The restriction imposed on the function $f$ This restricts, somewhat, but only superficially, the generality of the properties we want to establish. We will see, moreover, that the properties relating to the areas of the boundary of a convex set in the plane follow easily from this.

For a function ( $O$ ) the set of divided differences [ $x^{\prime},w^{\prime\prime};f$ ] For $x^{\prime},w^{\prime\prime}\in[a,b],\left(x^{\prime}\neq w^{\prime\prime}\right)$ is limited and we have

		$\displaystyle\inf\left[x^{\prime},x^{\prime\prime};f\right]=\lim\left[x^{\prime},x^{\prime\prime};f\right]=f_{a}^{\prime}(a),$
		$\displaystyle\sup\left[x^{\prime},x^{\prime\prime};f\right]=\lim\left[x^{\prime},x^{\prime\prime};f\right]=f_{g}^{\prime}(b).$

11.

Definition 1. The polygonal function $P\left(x_{0},x_{1},\ldots,w_{n};f\right)$ having its peaks ( $x_{0}=a,x_{n}=$ b) on the graph of the function $f:[a,b]\rightarrow\mathbb{R}$ is said to be registered in this function.

It's immediately clear that if $f$ is a function ( $d$ ) any polygonal function inscribed in this function is a non-concave function.

⁰⁰footnotetext: ^* Unilateral derivatives also exist, in the literal or improper sense, on

a

and on

b

We now have
Lemine 4. The half-lines supporting finite slopes $p,q$ , Or $p\leqq\leqq f_{a}^{\prime}(a)f_{\sigma}^{\prime}(b)\leqq q$ respectively at points ( $a,f(a)$ ), ( $b,f(b)$ ), cut off at a point $A(\alpha,\beta)$ whose absolution $\alpha$ is strictly included between a and $b$ .

The demonstration is immediate. We have

p\leqq f_{a}^{\prime}(a)<[a,b;f]<f_{a}^{\prime}(b)\leqq q

and then $\alpha$ , which is the only root of the equation in $x,-q(x-b)++p(x-a)=f(b)-f(a)$ is strictly between $a$ And $b$ Let us designate by $Q$ the polygonal function whose vertices are the points $(a,f(a)),A$ And ( $b,f(b)$ ) and either $P$ a polygonal function inscribed in the function $f$ The conditions of Lemma 3 are then verified and we have therefore $l(P)<l(Q)$ It follows that the length of inscribed polygonal functions is bounded above. This results in the well-known fact that the convex arc $y=f(x)$ , for ane function ( $O$ ), is rectifiable. Let us designate by $l(f)$ the length of this area, which we more simply call the length of the function $f$ We therefore have, by definition, $l(f)=\sup l(P),\mathscr{Q}_{i}$ being the set of polygonal lines inscribed in $\mathscr{G}_{i}$ the function $f$ Note that $l(f)$ is always a positive number. The same is true, moreover, of the length of a polygonal line.

We also have

l(f)=\lim_{(\mathscr{Q}i)}l(P)

(25)

since $f$ cannot coincide with a polygonal function inscribed in $f$ Note 1.
If $f$ is a function ( $C$ ), We have $l(P)<l(f)$ for everything $P\in\mathscr{Q}_{i}$ Indeed, if $P=P\left(x_{0},x_{1},\ldots,a_{n};f\right)$ and if we consider the point $\xi$ such as $w_{i}<\xi<w_{i+1}$ , for the polygonal function $P^{*}P^{*}\left(x_{0},x_{1},\ldots,x_{i},\quad\xi,x_{i+1},\ldots,x_{n};f\right)$ We have $l(P)<l\left(P^{*}\right)\leq l(f)$ , SO $l(P)<l(f)$ .

Note 2. In particular, $Q^{*}$ the polygonal line which is observed by taking $p=f_{a}^{\prime}(a),q=f_{\sigma}^{\prime}(b)$ It is then easy to see that the hypotheses of the lemma $3^{\prime}$ are verified by $Q$ And $Q^{*}$ and we deduce from this $l\left(Q^{*}\right)\leqq l(Q)$ equality occurs if, and only if, $Q$ coincides with $Q^{*}$ We will use this remark later.

The polygonal function $Q$ constructed above, satisfying condition 3 of Lemma 3 is a polygonal function consortia to the function $f$ We will further generalize this notion of a conscripted circular polygonal function.

Note 3. The length $l(f)$ of the arc of a function $(O)$ still possesses an additivity property. This property can be formulated in various ways and reduces to equality

l(f)=l(g)+l(h),

(26)

Or $f$ is a function ( $C$ ) defined on the interval $[a,b]$ And $g,h$ are its restrictions on subintervals $[a,e],[c,b]$ respectively where $a<c<b$ First, let us note that $g,h$ are also functions ( $C$ ).

So now $\varepsilon$ any positive number.
If $Q,R$ are polygonal functions inscribed in the functions $g,h$ such as $l(g)-\frac{\varepsilon}{2}<l(Q),l(h)-\frac{\varepsilon}{2}<l(R)$ , done $l(g)+l(h)-\varepsilon<<l(Q)+l(R)$ , if $P$ is the polygonal function extended on $[a,b]$ By formula (18) and taking into account (17), it follows that

l(g)+l(h)-\varepsilon<l(f).

(27)

On the other hand, either $P$ a polygonal function inscribed in the function $f$ such as $l(f)-\varepsilon<l(P)$ and either $P^{*}$ the inscribed polygonal function that we obtain from $P$ optionally adding the node $c$ (when it is not a knot of $P$ So if $Q^{*},R^{*}$ are the restrictions of $P^{*}$ on $[a,c]$ , $[c,b]$ respectively we have $l(f)-\varepsilon<l(P)\leqq l\left(P^{*}\right)=l\left(Q^{*}\right)+l\left(R^{*}\right)<<l(g)+l(h)$ , hence

l(f)-\varepsilon<l(g)+l(h).

(28)

From (27) and (28) it follows that equality (26), therefore precisely the additivity that we wanted to demonstrate.

Finally, note that Lemma 4 and the remarks that follow from it apply to any restriction of the function $f$ on a closed subset of $[a,b]$ 12.
We can now give the definition of a circumscribed polygonal function or line.

Definition 2. $f$ being a function $(C)$ a polygonal function $P$ having an odd number $2n+1\ (n>0)$ knots $a=x_{0}<x_{1}<\ldots<x_{2n}=b$ and whose corresponding vertices will be further designated by $P_{i},\ i=0,1,\ldots,2n$ , is said to be circumscribed to the function $f$ if the following two conditions are met:

1.

The peaks $P_{2i}$ corresponding to nodes with even index $\omega_{2i},\quad i==0,1,\ldots,n$ are on the curve $y=f(x)$ .
2.

Ohaque obté $P_{i}P_{i+1},i=0,1,\ldots,2n-1$ is a segment of a line supporting the function $f$ (of the curve $y=f(x)$ .)

Condition 1 implies $P\left(x_{2i}\right)=f\left(x_{2i}\right),\;i=0,1,\ldots,n$ The circumscribed polygonal function is therefore an interpolating function of the function $f$ , but only on the nodes $x_{2i},\;i=0,1,\ldots,n$ .

Condition 2 implies inequalities

	$\displaystyle{\left[\begin{array}[]{l}{\left[a,w_{1};P\right]\leqq f_{1}^{\prime}(a),f_{g}^{\prime}(b)\leqq\left[w_{2n-1},b;P\right]}\\ f_{o}^{\prime}\left(w_{2i}\right)\leqq\left[w_{2i-1},w_{2i};P\right],\left[w_{2i},w_{2i+1};P\right]\leqq f_{d}^{\prime}\left(w_{2i}\right)\end{array}\right.}$		(29)
	$\displaystyle i=1,2,\ldots,n-1(n>1)$

⁰⁰footnotetext: • The circumscribed polygonal function should not be confused

P

with the polygonal insert function

P\left(x_{0},x_{1},\ldots,x_{2n};f\right)

According to lemma 4 the knot $x_{2i-1}$ is well understood strictly between $x_{2i-2}$ And $x_{2i},i=1,2,\ldots,n$ We can also see that the peaks $P_{2i-1},i==1,2,\ldots,n$ are all below the function $f$ It follows that $P(x)<f(x)$ for everything $x$ different from a nowd of even index, which justifies the name for $P$ of polygonal function circumscribed to the function $f$ .

To simplify the language, we will say that the knots $x_{2i}$ and the respective peaks $P_{2i}$ nodes with even indices are of the first kind. The other nodes and vertices are of the second kind.
13. $f$ being a function (C), it follows that

f^{\prime}\left(w_{2i-2}\right)<f^{\prime}\left(w_{2i}\right),i=1,2,\ldots,n

(30)

From (29) it follows that

\left[x_{2i-2},x_{2i-1};P\right]<\left[x_{2i-1},x_{2i};P\right],i=1,2,\ldots,n.

(31)

However, it is important to note that if $n>1$ A circumscribed polynomial function is not necessarily a non-concave function. For this latter property to hold, it is necessary and sufficient that, apart from (30), the inequalities

\left[x_{2i-1},x_{2i};P\right]\leqq\left[x_{2i},x_{2i+1};P\right],i=1,2,\ldots,n-1

(32)

are verified.
Assuming always $n>1$ Among these polygonal functions, those for which equality holds in all relations (32) are particularly noteworthy. Then the sides $P_{2i-1}P_{2i},P_{2i}P_{2i+1}$ , form together for all $i$ a segment of a line of support at the point ( $\omega_{2i},f\left(\omega_{2i}\right)$ ), of the function $f$ .

It is easy to construct such circumscribed polygonal functions having any nodes of the first kind predetermined. Let us take, in fact, the points $x_{2i}$ such as $a=w_{0}<x_{2}<x_{4}<\ldots<x_{2n-2}<x_{2n}=b$ , but otherwise arbitrary and we conduct supporting lines $d_{i}$ at the points $P_{2i}\left(\omega_{2i},f\left(x_{2i}\right)\right),i=1,2,\ldots,n-1$ (if $n>1$ and are still $d_{0},d_{b}$ non-vertical support lines at points ( $a,f(a)$ ), ( $b,f(b)$ ).

So the slopes of the straight lines $d_{i}$ form an increasing sequence. The intersection $P_{2i-1}$ straight lines $d_{i-1},d_{i}$ has an abscissa $x_{2i-1}$ strictly between $x_{2i-2}$ And $x_{2i},\;i=1,2,\ldots,n$ The polygonal function with vertices $P_{i},\;i=0,1,\ldots,2n$ is indeed of the indicated form. It is easy to see that the polygonal function circumscribed at $f$ Thus constructed, it is indeed a non-concave function.

The preceding construction is based first on the fact that, if $f$ is a function $(C)$ , at each point of the graph of $f$ There exists at least one non-vertical support line. Then, the slope of a support line at the point $(x^{\prime},f(x^{\prime}))$ is always smaller than the slope of a support line at a point $(x^{\prime\prime},f(x^{\prime\prime}))$ having the abscissa $x^{\prime\prime}$ larger than $x^{\prime}$ .

14. The preceding construction helps to clarify the notion of consecutive circumscribed polygonal functions.

We will say that two polygonal functions $P,Q$ limited to the same function $f:[a,b]\rightarrow\mathbb{R}$ are consecutive (one to the other) if the sequence of first-kind knots of one of them is a partial sequence of the first-kind knots of the other. In this work, we are only interested in how one can deduce from a given circumscribed polygonal function $P$ whe certain other circumscribed polygonal function $Q$ , following $P$ .

We have
Lienme 5. If $P$ is a polygonal function circumscribed about the function $f$ , having the sequence of knots $\left(x_{i}\right)_{i=0}^{2n}$ , there is always another polygonal function $Q$ orconsumed to $f$ whose first-type knots form any increasing sequence $\left(\xi_{i}\right)_{i=0}^{n}\left(\xi_{0}=a,\xi_{n}=b\right)$ of which $\left(x_{2i}\right)_{i=0}^{n}$ is a partial sequence.

It suffices to show how a polygonal function can be constructed. $Q$ verifying the conditions of the stated lemma.

If $m=n$ we can take for $Q$ the polygonal function $P$ herself.
Suppose $m>n$ and either $\xi_{k_{i}}=x_{2i},i=0,1,\ldots,n$ . We have $k_{0}=0,k_{n}=m$ And $k_{0}<k_{1}<\ldots<k_{n}$ but at least one of the differences $k_{i}-k_{i-1}$ is greater than 1. If $k_{r}-k_{r-1}>1$ we modify the polygonal function $P$ in the meantime $\left[x_{2r-2},x_{2r}\right]$ by constructing the $\mathrm{n}^{0}$ previous. We then take [ $x_{2r-2},x_{2r}$ ] as an interval [ $a,b$ and we use lines of support at points with abscissas $\xi_{k_{r-1}+j},j=1,2$ , $\ldots,k_{r}-k_{r-1}-1$ and, to clarify the concepts, taking the slopes of the support lines at the extremities $x_{2r-2}$ And $x_{2r}$ respectively equal to $\left[x_{2r-2},x_{2r-1};P\right]$ And $\left[x_{2r-1},x_{2r};P\right]^{*}$ By making this construction for all the differences $k_{r}-k_{r-1}$ who are $>1$ We finally obtain a polygonal function $Q$ which satisfies the conditions of Lemma 5.

Note also that the polygonal function $Q$ thus constructed enjoys the property that if $P$ is a non-concave function, the same is true of $Q$ and we always $l(Q)<l(P)$ when $m>n$ .

We reiterate that we could study other polygonal functions circumscribed around the function $f$ and consecutive to $P$ , but the preceding will suffice for what follows.
15. Among all the polygonal leagues circumscribed about the function $f$ of the form $(C)$ , and having the same knots as the first type $x_{0},x_{2}$ , $\ldots,x_{2n}$ There is one, designated for the moment by $P^{*}$ , for which

\begin{gathered}{\left[x_{2i-2},x_{2i-1};P^{*}\right]=f_{d}^{\prime}\left(x_{2i-2}\right),\left[x_{2i-1},x_{2i};P^{*}\right]=f_{0}^{\prime}\left(w_{2i}\right)}\\ i=1,2,\ldots,n\end{gathered}

$\omega_{1},x_{3},\ldots,x_{2n-1}$ being (in increasing order of magnitude) the nodes of $P^{*}$ of the second kind. Since we have (30) ot also (if $n>1$ ), $f_{0}^{\prime}\left(x_{2i}\right)\leqq$

⁰⁰footnotetext: • This is not necessary, but allows for a precise definition of a polygonal function

Q

enjoying the desired properties.

$\leqq f_{a}^{\prime}\left(x_{i}\right),i=1,2,\ldots,n-1$ , we see that the circumscribed polygonal function $P^{*}$ is non-concave.

Finally, if we take into account remark 2 which follows lemma 4, we see that if $P$ is a circumscribed polygonal function having nodes of the first kind $x_{0},x_{2},\ldots,x_{2n}$ , We have $l\left(P^{*}\right)\leqq l(P)$ where equality only occurs if $P$ coincides with $P^{*}$ The polygonal function $P^{*}$ is therefore among all polygonal functions circumscribed about the function $f$ and which have the same knots of the second type $x_{0},x_{2},\ldots,x_{2n}$ , the one with the shortest length, and it is unique.
16. If $f$ is a function ( $C$ ), parallel to any non-vertical line there exists a support line which has a single point of contact with the function (with the curve $y=f(x)$ ) and the x-coordinate of this point belongs, of course, to the interval $[a,b]$ If the slope of the line is $\leqq f_{d}^{\prime}(a)$ this point of contact coincides with $a$ and if this angular coefficient is $\geqq f_{g}^{\prime}(b)$ it coincides with the end $b$ of the interval. Finally, if this angular coefficient is strictly between $f_{d}^{\prime}(a)$ And $f_{g}^{\prime}(b)$ , the point of contact is strictly between $a$ And $b$ .

We will now construct a circumscribed polynomial function in the following way.

Let us consider an increasing sequence $\left(p_{j}\right)_{j=0}^{n=0}(m>0)$ of real (finite) numbers in such a way that $p_{0}\leqq f_{a}^{\prime}(a)$ And $f_{\sigma}^{\prime}(b)\leqq p_{m}$ Let us designate by $d_{j}$ the line of support whose angular coefficient is $p_{j}$ and either $y_{j}$ the abscissa of the point of contact of $d_{j}$ with the curve $y=f(x)$ The sequel $\left(y_{j}\right)_{j=0}^{n!}$ is non-decreasing and, by construction, we have $y_{0}=a,y_{m}=b$ Among the points $y_{j}$ he $y$ has at least 2 distinct ones (the points $y_{0}$ And $y_{m}$ (in any case). Let us then designate by ( $a=$ ) $x_{0},x_{2},\ldots,a_{2n}(=b)$ the growing series of bridges of $[a,b]$ with which the points coincide $y_{j}$ For everything $j$ there is a $i$ such as $y_{j}=x_{2i}$ and for everything $i$ at least one $j$ such as $x_{2i}=y_{j}$ Let us then designate by $s_{i}$ the number of points $y_{j}$ which coincide with $x_{2i}$ . SO $s_{i}$ are positive and their sum is equal to $m+1$ . We have $y_{k}=x_{2i}$ For $s_{0}+s_{1}+\ldots s_{i-1}\leqq k\leqq s_{0}+s_{1}+\ldots+s_{i}-1,i=0,1,\ldots,n\left(s_{0}+\right.+s_{1}+\ldots s_{i-1}$ being replaced by 0 if $i=0$ ). To simplify the writing, let's add

\left\{\begin{array}[]{c}d_{0}^{\prime}=d_{s_{0}-1},d_{2n-1}^{\prime}=d_{s_{0}+s_{1}+\ldots+s_{n-1}}\\ d_{2i-1}^{\prime}=d_{s_{0}+s_{1}+\ldots+s_{i-1}},d_{2i}^{\prime}=d_{s_{0}+s_{1}\ldots+s_{i-1}}\\ i=1,2,\ldots,n-1\end{array}\right.

(where we only retain the first two equalities if $n=1$ ).
SO $d_{0}^{\prime},d_{2,5-1}^{\prime}$ are lines of support respectively at the points ( $a,f(a)$ ), ( $b,f(b)$ ) And $d_{2i-1}^{\prime},d_{2i}^{\prime}$ are both lines of support at a point ( $x_{2i},f\left(x_{2i}\right)$ ), $i=1,2,\ldots,n-1$ (when $n>1$ ). Let us also designate by $P_{2i}$ the point $\left(x_{2t},f\left(x_{2t}\right)\right),i=0,1,\ldots,n$ and by $p_{2i-1}$ the intersection of the lines of support $d_{2i-2}^{\prime},d_{2i-1}^{\prime},i=1,2,\ldots,n$ So, according to definition 2, the polygonal function $P$ having as vertices the points $P_{i},i=0,1,\ldots$ , $\ldots,2n$ and whose nodes form the increasing sequence ( $\left.w_{i}\right)_{i=0}^{2n}$ , is circumscribed
to the function $f$ and is indeed a non-concave function. Note that for this polygonal function we have

\begin{gathered}0<\left[w_{2i-1},w_{2i};P\right]-\left[w_{2i-2},w_{2i-1};P\right]=\\ =p_{s_{0}+s_{1}+\ldots+s_{i-1}}-p_{s_{0}+s_{1}+\ldots+s_{i-1}-1},i=1,2,\ldots,n\end{gathered}

from which it also follows that

	$\displaystyle 0<\left[x_{2i-1},x_{2i};P\right]-\left[x_{2i-2},x_{2i-1};P\right]\leqq\max_{j=1,2,\ldots,m}\left(p_{1}-p_{j-1}\right)$		(34)
	$\displaystyle i=1,2,\ldots,n.$

17. We can now demonstrate the*

Article 6. Given any two positive numbers $\varepsilon_{1},\varepsilon_{2}$ , one can always find a polygonal function $P$ oirconsorite $\grave{a}$ the function $f$ (which is a function $(C)$ ) and whose nodes form the increasing sequence $\left(x_{i}\right)_{i=0}^{2n}$ , so that the following two conditions are met:

1.

We have

0<\left[x_{2i-1},x_{2i};P\right]-\left[x_{2i-2},x_{2i-1};P\right]<\varepsilon_{1},i=1,2,\ldots,n

(35)

2.

The difference $x_{j}-x_{j-1}$ of two consecutive knots is $<\varepsilon_{2}$ Therefore, we have inequality.

\max_{j=1,2,\ldots,2n}\left(x_{j}-x_{j-1}\right)<\varepsilon_{2}.

(36)

By slightly modifying the notation for the purposes of the demonstration, we see that, according to the construction that led us to the inequalities (34), we can first construct a circumscribed polygonal fonotion $P^{\prime}$ , whose sequence of nodes is $\left(w_{i}^{\prime}\right)_{i=0}^{2k}$ and for which

0<\left[x_{2i-1}^{\prime},x_{2i}^{\prime};P^{\prime}\right]-\left[x_{2i-2}^{\prime},x_{2i-1}^{\prime};P^{\prime}\right]<\varepsilon_{1},i=1,2,\ldots,k.

If now $x_{j+1}^{\prime}-x_{j}^{\prime}<\varepsilon_{1},j=0,1,\ldots,2k-1$ it is enough to take as $P$ this polygonal function $P^{\prime}$ and conditions (35), (36) of Lemma 6 are verified.

Otherwise, according to Lemma 5, we can construct the circumscribed polygonal line $P$ consecutive to $P^{\prime}$ so that conditions (35), (36) are satisfied. To achieve this result, it suffices to insert between two consecutive nodes of the first species $x_{2i-2}^{t},x_{2i}^{\prime}$ of $P^{\prime}$ a sufficient and suitably distributed number of knots of the first type of $P$ For example, we can insert between $x_{2i-2}^{\prime}$ And $x_{2i}^{\prime}$ such as knots of

⁰⁰footnotetext: • It is to realize the conditions of this lemma that we initially limited ourselves to functions (

C

the first species the points that divide the interval [ $x_{2i-2}^{\prime},x_{2i}^{\prime}$ ] in $r$ equal parts, $r$ being a natural number $>\frac{x_{2i}^{t}-x_{2i-2}^{t}}{\varepsilon_{2}}$ and by constructing this structure, $i=1,2,\ldots,k$ .

Lemma 6 is thus proven.
18. We can now prove:

Theorem 1. If $f$ is a function ( $C$ ) And $\mathscr{Q}_{e}$ is the set of polygonal functions circumscribed to $f$ we have

\inf_{P\in\mathscr{L}}l(P)=l(f)

(37)

We will demonstrate this in two steps.
First step. We will first demonstrate that

\inf_{P\in Q_{c}}l(P)\geqq l(f).

(38)

Either $P\in\mathbb{2}_{c}$ And $\left(o_{2i}\right)_{i=0}^{n}$ the sequence of nodes of the first species of $P$ . If $\varepsilon$ is any positive number, we can construct an increasing sequence $\left(y_{j}\right)_{j=0}^{m}$ , Or $y_{0}=a,y_{m}=b$ , of which $\left(x_{2i}\right)_{i=0}^{n}$ is a partial sequence such that if $P^{(i)}$ is the restriction on the interval $\left[x_{2i-2},x_{2i}\right]$ of the inscribed polygonal line $P^{*}\left(y_{0},y_{1},\ldots,y_{m};f\right)$ and if $f^{(i)}$ is the restriction on the same interval of $f$ , we have

l\left(f^{(i)}\right)--\frac{\varepsilon}{n}<l\left(P^{(i)}\right),i=1,2,\ldots,n,

(39)

This results from the definition of the length of a function (C): Given the additivity property of the lengths of polygonal functions and of the functions ( $O$ ), we have

\sum_{i=1}^{n}l\left(f^{(i)}\right)=l(f),\sum_{i=1}^{n}l\left(P^{(i)}\right)=l\left(P^{*}\right).

From (39) it then follows $l(f)-\varepsilon<\left(P^{*}\right)$ But, by applying Lemma 3, we have $l\left(P^{*}\right)<l(P)$ and it follows that $l(f)-\varepsilon<l(P)$ , hence inequality (38).

Second step. We will demonstrate that

\inf_{P\in\mathscr{Q}_{c}}l(P)\leqq l(f).

(40)

Let e be any positive number and let $P$ a polygonal function circumscribed to $f$ verifying condition 1 of the lemma with $\varepsilon_{1}==\frac{2\varepsilon}{u(f)}\cdot\left(x_{i}\right)_{i=0}^{2n}$ being the sequence of nodes of $P$ , let us designate by $P*$ the inscribed polygonal function $P*\left(x_{0},x_{2},\ldots,x_{2n};f\right)$ has $f$ Therefore, we can apply inequality (11) to triangles. $P_{2i-2}P_{2i-1}P_{2i};i=1,2,\ldots$ , or the letters $P_{i}$
always designate the vertices of $P$ Taking into account the additivity of length, we obtain

\begin{gathered}l(P)<\left(1+\frac{\varepsilon_{1}}{2}\right)l\left(P^{*}\right)=\left(1+\frac{\varepsilon}{l(f)}\right)l\left(P^{*}\right)<\\ <\left(1+\frac{\varepsilon}{l(f)}\right)l(f)=l(f)+\varepsilon\end{gathered}

and inequality (40) follows.
Theorem 1 is proven.
It is easy to see, by applying Lemma 3, that if $f$ is a function $(C)$ we have $l(f)<l(P)$ for all $P\in\mathscr{E}_{c}$ We can therefore
deduce that

\lim_{\bar{P}\in\mathscr{Q},}l(P)=l(f)

(41)

§ 4. ON SOME APPROXIMATE THIRD-CENTURIES

19.

Either $\omega(\delta)$ the oscillation modulus of the function $f:[\alpha,b]\rightarrow\mathbb{R}\omega(\delta)$ is defined for all $\delta>0$ and we have $\left|f\left(\omega^{\prime}\right)-f\left(\omega^{\prime\prime}\right)\right|\leqq\omega\left(\left|\omega^{\prime}-\omega^{\prime\prime}\right|\right)$ if $x^{\prime}\neq x^{\prime\prime}$ Among the various properties of $\omega(\delta)$ Let us remember that it is a non-decreasing function and that we have $\lim_{\delta\downarrow 0}\omega(\delta)=0$ if and only if $f$ is continuous, so in particular if $f$ is a function ( $O$ ). This last property also expresses the uniform continuity of the function $f$ .

The first approximation theorem is expressed by Theorem
2. If $f:[a,b]\rightarrow\mathbb{R}$ is a function ( $C$ and if a is any positive number, we can always find a polygonal function $P=P\left(x_{0},x_{1},\ldots,x_{n};f\right)$ inserts into $f$ such that one has

|f(x)-P(x)|<\varepsilon\text{ pour }x\in[a,b].

(42)

The property is well known. To be thorough, we will provide the proof. The restriction of $P$ on the partial interval [ $x_{4-1},x_{i}$ ] coincides with the first-degree polynomial $\frac{1}{x_{i}-x_{i-1}}\left[\left(x-x_{i-1}\right)f\left(x_{i}\right)+\right.\left.+\left(x_{i}-x\right)f\left(x_{i-1}\right)\right]$ and no, let's deduce

\begin{gathered}|f(x)-P(x)|\leqq\\ \leqq\frac{1}{x_{i}-x_{i-1}}\left[\left(x-x_{i-1}\right)\left|f(x)-f\left(x_{i}\right)\right|+\left(x_{i}-x\right)\left|f(x)-f\left(x_{i-1}\right)\right|\right]\leqq\\ \leqq\frac{1}{x-x_{i-1}}\left[\left(x-x_{i-1}\right)\omega\left(x_{i}-x\right)+\left(x_{i}-x\right)\omega\left(x-x_{i-1}\right)\right]\\ \leqq\omega\left(x_{l}-x_{i-1}\right)\text{ pour }x\in\left[x_{i-1},x_{i}\right]\end{gathered}

These inequalities result from noting that for $x\in\left[x_{i-1},x_{i}\right]$ we have $0\leqq x_{i}-x,x-x_{i-1}\leqq x_{i}-x_{i-1}$ .

So now

\eta=\max_{i=1_{2}^{2},\ldots,n}\left(x_{i}-x_{i-1}\right).

It follows that

|f(x)-P(x)|\leqq\omega(\eta)\text{ pour }x\in[a,b].

By choosing the nodes $x_{i}$ so that $\omega(\eta)<\varepsilon$ , we obtain inequality (42). The choice of points $x_{i}$ can be done, for example, by taking the points that divide into $n$ equal parts the interval $[a,b]$ , $n$ being a sufficiently large natural number.

20. Moreover, we know that the property expressed by Theorem 2 is true for any continuous function $f:[a,b]\rightarrow\mathbb{R}$ But if $f$ is a function $(C)^{\star}$ We can obtain a more complete result. This result is expressed by the

Theorem 3. If $f:[a,b]\rightarrow\mathbb{R}$ is a function ( 0 ) and if e is any positive number, we can always find a polygonal function $Q$ registered in $f$ , such as one might have

0\leqq Q(x)-f(x)<\varepsilon\text{ pour }x\in[a,b]

(43)

And

0<l(f)-l(Q)<\varepsilon.

(44)

The demonstration presents no difficulties. Let $P$ the polygonal function inscribed in $f$ which satisfies condition (42). Let $Q$ a polygonal function inscribed in $f$ which we deduce from $P$ by adding a number of new nodes, outside of the existing nodes $x_{i}$ of $P$ We can also say that $Q$ is a consecutive polygonal function $\grave{a}P$ , this time registered in $f$ The continuation of the knots of $P$ is a partial sequence of the sequence of nodes of $Q^{**}$ .

Now, regarding the definition of the function's length $f$ It follows that we can choose these new nodes in such a way that if $f^{(i)}$ And $Q^{(i)}$ are respectively the restrictions of $f$ And $Q$ on the interval $\left[\mathscr{W}_{i-1},\mathscr{x}_{i}\right]$ , we have

0<l\left(f^{(i)}\right)-l\left(Q^{(i)}\right)<\frac{\varepsilon}{n},i=1,2,\ldots,n.

The additivity property of length then shows us that we have (44). As for inequalities (43), they result from the fact that $\forall_{i}Q(x)\leqq\leqq P(x)$ , done $\forall_{x}0\leqq Q(x)-f(x)\leqq P(x)-f(x)$ .

⁰⁰footnotetext: • or if

f

has a somewhat more general structure, as we will see in §

s^{u\text{ lvant. }}

** Inscribed polygonal function

Q

is therefore obtained, as in the case of circumscribed and consecutive polygonal functions, by a refinement of the sequence of nodes of

P

We can also see that we can find an infinite sequence. $\left(P^{(n)}\right)_{n=0}^{\infty}$ non-increasing* of inscribed polygonal functions, converging uniformly to the function $f$ (on $[a,b]$ ) and, at the same time, the increasing sequence of lengths $\left(l\left(P^{(n)}\right)_{n=0}^{\infty}\right.$ converging towards $l(f)$ We can leave the demonstration of this property to the reader.
21. We will establish an analogous property for polygonal functions circumscribed to $f$ .

Theorem 4. If $f:[a,b]\rightarrow\mathbb{R}$ is a function $(C)$ and if e is any positive number, we can find a polygonal function $P$ confined to $f$ , such as one might have

|f(x)-P(x)|<\varepsilon\text{ pour }x\in[a,b].

(45)

For the demonstration, we will rely on Lemmas 1 and 6. Let's choose positive numbers. $\varepsilon_{1}$ , $\varepsilon_{2}$ such as

\varepsilon_{1}\leqq\sqrt{2},\varepsilon_{2}<\frac{\varepsilon}{2\sqrt{2}},\omega\left(2\varepsilon_{2}\right)<\frac{\varepsilon}{\sqrt{2}}

(46)

which is possible according to the properties of the oscillation modulus $\omega(\delta)$ of the function $f$ .

So now $P$ the polygonal function circumscribed about the function $f$ , having the nodes $x_{i},i=0,1,\ldots,2n$ and which satisfies conditions (35) and (36) of Lemma 6. To evaluate the difference $f(x)-P(x)$ Let us consider the triangle formed by the vertices $B,A,C$ of $P$ of respective x-coordinates $x_{2i-2},x_{2i-1},x_{2i}$ Let us designate by $D,F$ the intersections of the vertical of the point $x$ of $\left[x_{2i-2},x_{2i}\right]$ with the sides $AC,BC$ respectively. We then have $0\leqq f(x)-P(x)\leqq\overline{JF},\bar{K}\bar{F}$ being the length of the line segment $EF^{\prime}$ But, obviously (compare also with figure 2), $\overline{EF}<$ d. Applying Lemma 1, it follows that

	$\displaystyle 0\leqq f(x)-P(x)\leqq\mathrm{d}$	$\displaystyle<\sqrt{\left(f\left(x_{2i}\right)-f\left(x_{2i-2}\right)\right)^{2}+\left(x_{2i}-x_{2i-2}\right)^{2}}\leqq$
		$\displaystyle\leqq\sqrt{\omega^{2}\left(2\varepsilon_{2}\right)+4\varepsilon_{2}^{2}<\varepsilon}.$

We can therefore deduce that

|f(x)-P(x)|<\varepsilon,\text{ pour }x\in\left[x_{2i-2},x_{2i}\right],\quad i=1,2,\ldots,n

and inequality (45) is proven.
22. As with inscribed polygonal functions, a more complete result can be obtained. This result is expressed by the

\text{ * donc }\forall n\forall xP^{(n+1)}(x)\leqq P^{(n)}(x)\text{. }

Theorimiz 5. If $f:[a,b]\rightarrow\mathbb{R}$ is a function ( $C$ ) and if $\varepsilon$ is any positive number, we can always find a polygonal function $Q$ circumscribed to $f$ , such as one might have

0\leqq f(x)-Q(x)<\varepsilon\text{ pour }x\in[a,b]

(47)

And

0<l(Q)-l(f)<\varepsilon.

(48)

The proof presents no difficulties, taking into account Theorem 1. Starting from the polygonal function $P$ which satisfies condition (45) of Theorem 4, we can find a circumscribed polygonal function $Q$ consecutive to $P$ such that we have (48). Since, by construction, we have $\forall_{x}P(x)\leqq Q(x)$ , SO $\forall_{x}0\leqq f(x)-Q(x)\leqq f(x)-P(x)$ , we also deduce inequality (47).

Finally, we can see that we can find an infinite sequence. $\left(P^{(n)}\right)_{n=0}^{\infty}$ non-decreasing * of circumscribed polygonal functions ò $f$ converging uniformly towards the function $f$ (on $[a,b]$ ) and at the same time the decreasing sequence of lengths ( $\left.lI^{(n)}\right)_{n=0}^{\infty}$ converging towards $l(f)$ We can leave the demonstration of this property to the reader.

It is clear that we cannot extend Theorem 4 to a continuous function here. $f$ arbitrary. Indeed, we have not defined the notion of a polygonal function circumscribed about such a function.

Section 5. Final Questions

23.

Several of the previous results can be extended by lifting certain restrictions to which the function is subject. $f$ or by generalizing this function. We can also, in a certain sense, generalize the notion of a circumscribed polygonal function.

In what follows, we will briefly examine these questions without dwelling too much on the proofs.
24. Instead of a function $(C)$ consider an arbitrary convex and continuous function defined on $[a,b]$ In this case, the unilateral derivatives $f_{d}^{\prime}(a),f_{d}^{\prime}(b)$ can be infinite, the first one being equal to $-\infty$ and the second to $+\infty$ There is nothing to say about polygonal functions embedded within the function, as definition 1 still retains a precise meaning, but there is some difficulty in defining a polygonal function circumscribed about the function, as definition 2 does not always have a precise meaning. Indeed, if, for example, $f_{d}^{\prime}(a)=-\infty$ Or $f_{\sigma}^{\prime}(b)=+\infty$ , the only line of support that can be drawn to the point ( $a,f(a)$ ) out ( $b,f(b)$ ) is vertical. We can circumvent the difficulty by admitting that a polygonal line can also have vertical extreme sides. More precisely, we can agree that in the following $\left(x_{i}\right)_{i=0}^{\infty}$ nodes we can have $x_{0}=x_{1}$ the peaks $P_{0},P_{1}$ not being confused and the vertical segment $P_{0}P_{1}$ having a non-zero length. Similarly, we can have $x_{n-1}=x_{n}$ the peaks $P_{n-1},P_{n}$ not being confused and the vertical segment $P_{n-1}P_{n}$ having a non-zero length ^⋆⋆ . In any case

⁰⁰footnotetext: • therefore

\forall_{n}\forall_{x}:P^{(n)}(x)\leqq P_{u^{+}}{}^{+1)}(x)

.
** Or we can also introduce other polygonal lines having vertical sides, but they will not interfere with the work.

if $n>2$ the sequel $\left(w_{i}\right)_{i=1}^{n-1}$ is increasing. We can then maintain definition 2 of a circumscribed polygonal line, noting that it is no longer, in general, a function. Indeed, by definition, a function can only take one value for any value of the variable, which, for the polygonal lines considered, may not occur at the points $a$ And $b$ The polygonal lines thus defined still possess the properties established in § 4. In particular, they have a well-defined length which is $>l(f)$ 25.
We can still circumvent the difficulty by replacing definition 2 with the

DEFINITION 3. If $f:[a,b]\rightarrow\mathbb{R}$ is a function ( $C$ ), the polygonal function $P$ having the increasing sequence of nauds $\left(x_{i}\right)_{i=1}^{2n}$ Or $x_{0}=a,x_{2n}=b$ , is said to be circumscribed to $f$ if the sides $P_{2i-2}P_{2i-1}$ And $P_{2i-1}P_{2i}$ are segments of support lines at points ( $\left.\omega_{2i-1},f\left(\omega_{2i-1}\right)\right)$ For $i=1,2,\ldots,n$
It follows from the definition that

P(a)\leqq f(a),P(b)\leqq f(b).

Therefore, definition 3 applies even if $f$ is only continuous and convex (without having bounded one-sided derivatives).

The property $l(P)>l(f)$ subsist ot the theorems $1,4,5$ are still valid.

Another way of turning the difficulties around results from this

26.

We can also generalize the problem by assuming only that $f:[a,b]\rightarrow\mathbb{R}$ is a convex function. It may therefore not be continuous at the endpoints $a$ And $b$ In this case, the function $g:[a,b]\rightarrow\mathbb{R}$ is defined by the formulas

g(x)=\left\{\begin{array}[]{l}f(a+0)\text{ pour }x=a,\\ f(x)\text{ pour }x\in]a,b[,\\ f(b-0)\text{ pour }a=b,\end{array}\right.

is a continuous convex function.
To maintain the previous properties, the simplest approach is to always complete the polygonal lines inscribed and circumscribed in and to $g$ by vertical sides at the points $a$ And $b$ of lengths (which can also be zero) respectively equal to $f(a)-f(a+0)$ And $f(b)-f(b-0)$ 27.
The preceding results can be partly extended to non-concave functions defined on a bounded and closed interval $[a,b]$ The definitions of embedded and circumscribed polygonal lines are done in the same way. The properties remain, noting only that equality $l(P)=l(f)$ can well occur both for an inscribed polygonal line and for a circumscribed polygonal league $f$ We can first consider only continuous non-concave functions with bounded derivatives and then generalize the properties, as we did for the functions ( $C$ ).

Formula (25) may not be true. Indeed, if the function $f$ reduces to a polynomial of degree 1, any polygonal function inscribed in $f$ , in accordance with definition 1, coincides with $f$ and we have done $l(P)==l(f)$ whatever $P\in\mathscr{2}_{i}$ Moreover, it can be shown that outside of this case formula (25) is true.

On the contrary, formula (41) is always valid. To show us that this is so, it suffices to consider the case where $f$ reduces to a polynomial of degree 1. In this case, any polygonal function $P$ having the (wood) knots $a,\frac{a+b}{2},b$ and for which $\lambda=f\left(\frac{a+b}{2}\right)-P\left(\frac{a+b}{2}\right)\geqq 0$ , $f(a)=P(a),f(b)=P(b)$ is limited to $f$ and we have (for $\lambda>0$ ) $l(f)<l(P)$ , as well as lim $l(P)=l(f)$ For $\lambda\nmid 0$ 28.
Before going further, let us note that we can also generalize the notion of a polygonal line or circumscribed polygonal function by imposing, instead of (28), only the conditions

\begin{gathered}{\left[x_{2i-2},x_{2i-1};P\right]\leqq f_{n}^{\prime}\left(x_{2i-2}\right),f_{n}^{\prime}\left(x_{2i}\right)\leqq\left[x_{2i-1},x_{2i};P\right]}\\ i=1,2,\ldots,n\end{gathered}

Such a polygonal function may not be a non-concave function. It is of little interest since one can always find another circumscribed polygonal function with at most the same length, which is non-concave and has the same nodes of the first kind.
29. To justify the proof proposed by G. Tzitzéica for inequality (1), it suffices to put the property expressed by Lemma 3 into a more general form.

If the length $l(f)$ of the non-concave function $f:[a,b]\rightarrow\mathbb{R}$ is defined by $l(f)=l(g)+f(a)-f(a+0)+f(b)-f(b-0)$ , Or $g$ is the function (49), we have the

Theorem 6. If $f$ , $\varphi$ are two non-concave functions defined on the bounded and closed interval $[a,b](a<b)$ and if the following conditions are met:

1.

$f(a)=\varphi(a),f(b)=\varphi(b)$ ;
2.

on $a\forall_{x}f(x)\leqq\varphi(x)$ ,
then between the lengths of the functions $f,\varphi$ we have inequality

l(\varphi)\leqq l(f),

(50)

equality is only true if the functions $f$ And $\varphi$ coincide.

For the demonstration, it suffices first to assume that $f$ And $\varphi$ are functions $(C)$ Then, it is easy to extend the property to any non-concave functions.

Suppose then that $f,\varphi$ are functions ( $C$ ). Either $P$ a polygonal function inscribed in $\varphi$ And $Q$ a non-concave polygonal function circumscribed at $f$ Polygonal functions $P,Q$ verify the conditions of the lemma $3^{\prime}$ and we have done $l(P)\leqq l(Q)$ , hence sup $l(P)\leqq$ inf $l(Q)$ And $(50)$ results from Theorem 1.

Let us now suppose that $x_{0}$ is a point of $]a,b\left[\right.$ such as $f\left(x_{0}\right)<\varphi\left(x_{0}\right)$ A line of support to $\varphi$ to the point ( $x_{0},\varphi\left(x_{0}\right)$ ) cuts the curve $y=f(x)$ at the points with abscissas $a^{\prime}$ And $b^{\prime}$ Or $a\leqq a^{\prime}<b^{\prime}\leqq b$ Consider the function defined by the tormules

h(x)=\left\{\begin{array}[]{l}f(x)\text{ pour }x\in\left[a,a^{\prime}\right]\cup\left[b^{\prime},b\right],\\ P\left(a^{\prime},b^{\prime};f\right)\text{ pour }x\in\left[a^{\prime},b^{\prime}\right].\end{array}\right.

SO $h$ is a non-concave function and we have $\forall_{x}f(x)\leqq h(x)\leqq\varphi(x)$ , SO $l(\varphi)\leqq l(h)\leqq l(f)$ But the additivity of length shows us that $l(h)<l(f)$ We can therefore deduce that $l(\varphi)<l(f)$ 30.
Now let us consider, in the plane, the boundary $\Gamma$ of a bounded and convex set. T is a closed convex curve. Let us first assume that every line of support intersects $\Gamma$ at a single point. Let then be on $\Gamma$ three points $A,B,C$ to which we can draw lines of support forming a non-degenerate triangle. Then each of the convex arcs $AB,BC,CA$ is a graphical representation of a function ( $O$ ) with respect to two suitable coordinate axes. This allows us to define polygonal lines circumscribed to $\Gamma$ , by connecting polygonal lines circumscribed about the arcs $AB,BC,CA$ We thus arrive at defining polygons circumscribed about $\Gamma$ by varying the points $A,B,C$ on $\Gamma$ It is then clear what is meant by the perimeter (the length). $l(\Gamma)$ of $\Gamma$ and the perimeter $l(P)$ of a polygon $P$ confined to $\Gamma$ .

This definition can be extended to convex curves. $\Gamma$ containing also straight sections. We still have the inequality

l(\Gamma)\leqq l(P)

(51)

$P$ being a polygon circumscribed to $\Gamma$ Inequality (51) is ,
moreover, a special case of a more general property which will be noted later.
31. It can be shown that if in the plane, $\Gamma,\Gamma_{1}$ are the boundaries of two bounded convex sets, the second of which contains the first, we have

l(\Gamma)\leqq l\left(\Gamma_{1}\right)

(52)

Without needing to consider the general case, let us simply assume that the curve $\Gamma$ is completely internal to $\Gamma_{1}$ So then $A,B$ two points on 1 to which parallel lines of support can be drawn and are: $A^{\prime},A^{\prime\prime}$ , respectively $B^{\prime},B^{\prime\prime}$ the points where these lines of support intersect the curve $\Gamma_{1}$ Let us also designate by $a^{\prime},a^{\prime\prime},b^{\prime},b^{\prime\prime}$ the lengths of the line segments $A^{\prime}A,A^{\prime\prime}A,B^{\prime}B$ , $B^{\prime\prime}B$ , by $l_{1},l_{2}$ the lengths of the ares $AmB$ , $AnB$ of $\Gamma$ and by $L_{1},L_{2},L_{3},L_{4}$ the lengths of the ares $A^{\prime}\alpha A^{\prime\prime},A^{\prime\prime}\beta B^{\prime},B^{\prime\prime}\gamma B^{\prime},B^{\prime}\delta A^{\prime}$ of $\Gamma_{1}$ (see figure 8). Given the additivity of length, we have $l(\Gamma)=l_{1}+l_{2}$ , $l\left(\Gamma_{1}\right)=L_{1}+L_{2}+L_{3}+L_{4}$ However, according to the previous results, we have $a^{\prime}+a^{\prime\prime}<L_{1}$ ,

$b^{\prime}+b^{\prime\prime}<L_{3},l_{1}<a^{\prime\prime}+L_{2}+b^{\prime\prime},l_{2}<a^{\prime}+L_{4}+b^{\prime}$ , hence $l(\Gamma)<l\left(\Gamma_{1}\right)$ , therefore inequality (52).

It is easy to show that in (52) equality is only possible if $\Gamma$ And $\Gamma_{1}$ coincide.

HIBLIOGRAPHY

1.

Hille, Einar, Analysis, $i^{6F}$ vol., 1064.
2.

Mitrinovio, DS, Analytic inequalities, 1970.
3.

Popovidu, T., Notes on the conventional functions of superior order (IX). Bull. Math. de la soc. Roumaniene des Sei,, 1942, 43, 85-141.
4.

Popoviciu, T., Curs de analiză matematică, III ^e part., 1974.
5.

Taitzerca, G., O proprietale a sinusului. Gazeta matematică, 1912/13, 18, 407.

Polygonal lines inscribed and circumscribed to a convex arc

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Paper (preprint) in HTML form

POLYGONAL LINES INSCRIBED AND CIRCUMSCRIBED WITH A CONVEX ARC

Section 1. Preliminary Questions

§ 2. SOME PROPERTIES OF POLYGONAL FUNCTIONS

§ 3. ON THE LENGTH OF INSCRIBED OR CIRCUMCROACHED POLYGONAL LINES $\mathring{\mathrm{A}}$ A CONVEX SILVER

17. We can now demonstrate the*

§ 4. ON SOME APPROXIMATE THIRD-CENTURIES

Section 5. Final Questions

HIBLIOGRAPHY

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Polygonal lines inscribed and circumscribed to a convex arc

Abstract

Authors

Original title (in French)

Keywords

Cite this paper as

PDF

About this paper

Journal

Publisher Name

DOI

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Paper (preprint) in HTML form

POLYGONAL LINES INSCRIBED AND CIRCUMSCRIBED WITH A CONVEX ARC

Section 1. Preliminary Questions

§ 2. SOME PROPERTIES OF POLYGONAL FUNCTIONS

§ 3. ON THE LENGTH OF INSCRIBED OR CIRCUMCROACHED POLYGONAL LINESHAS̊\mathring{\mathrm{A}}A CONVEX SILVER

17. We can now demonstrate the*

§ 4. ON SOME APPROXIMATE THIRD-CENTURIES

Section 5. Final Questions

HIBLIOGRAPHY

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§ 3. ON THE LENGTH OF INSCRIBED OR CIRCUMCROACHED POLYGONAL LINES $\mathring{\mathrm{A}}$ A CONVEX SILVER