Notes on generalizations of higher-order convex functions (II)

Tiberiu Popoviciu
(original), Approximation Theory, convexity, Mathematical Analysis, paper

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T. Popoviciu, Notes sur les généralisations des fonctions convexes d’ordre supérieur (II), Bull. de la Sect. Sci. de l’Acad. Roum., 22 (1940) no. 10, pp. 473-477 (in French).

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1940 d -Popoviciu- Bull. Sect. Sci. Acad. Roum. – Notes sur les generalisations des fonctions convexes d’ordre superieur (II)

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Bulletin de la Section Scientifique de l’Académie Roumaine

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[MR0003435, JFM 66.0241.02]

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1940 d -Popoviciu- Bull. Sect. Sci. Acad. Roum. - Notes on the generalizations of convex functions
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ROMANIAN ACADEMY

BULLETIN OF THE SCIENTIFIC SECTION

NOTES ON GENERALISATIONS OF HIGHER ORDER CONVEX FUNCTIONS (II)

BY

TIBERIU POPOVICIUNote presented by Mr. S. Stoïlow, Mc.AR in the session of May 17, 1940

ORDER FUNCTIONS n n nnnBY SEGMENTS

I. In the previous note we considered the order functions ( n ∣ k n ∣ k n∣kn \mid kn∣k), whose order functions n n nnnare a special case ( k = 0 k = 0 k=0k=0k=0). We will now study another class of functions, closely related to these functions.
We will say that the linear set E E EEEis decomposed into m m mmmconsecutive subsets E 1 , E 2 , … , E m E 1 , E 2 , … , E m E_(1),E_(2),dots,E_(m)E_{1}, E_{2}, \ldots, E_{m}E1,E2,…,Emif: 1 ∘ E i ⊆ E , i = 1 , 2 , … , m , 2 ∘ 1 ∘ E i ⊆ E , i = 1 , 2 , … , m , 2 ∘ 1^(@)E_(i)sube E,i=1,2,dots,m,2^(@)1^{\circ} E_{i} \subseteq E, i=1,2, \ldots, m, 2^{\circ}1∘Ei⊆E,i=1,2,…,m,2∘any point of E E EEEbelongs to one of the E i , 3 ∘ E i , 3 ∘ E_(i),3^(@)E_{i}, 3^{\circ}Ei,3∘any point of E i E i E_(i)E_{i}Eiis to the left of any point of E i + 1 , i = 1 , 2 , … , m − 1 E i + 1 , i = 1 , 2 , … , m − 1 E_(i+1),i=1,2,dots,m-1E_{i+1}, i=1,2, \ldots, m-1Ei+1,i=1,2,…,m−1. The subsets E i E i E_(i)E_{i}Eiof E E EEEare therefore disjoint sections of E E EEEand completely exhausting the whole thing E E EEE.
Let us now introduce
Definition I. We will say that the function f f fffis in order n n nnnby segments on E E EEEif we can decompose the whole thing E E EEEin a finite number m m mmmof consecutive subsets
(I) E 1 , E 2 , … , E m , (I) E 1 , E 2 , … , E m , {:(I)E_(1)","E_(2)","dots","E_(m)",":}\begin{equation*} E_{1}, E_{2}, \ldots, E_{m}, \tag{I} \end{equation*}(I)E1,E2,…,Em,
such that on each of these subsets the function is of order n n nnn.
In particular, for n = 0 n = 0 n=0n=0n=0, we can say that f f fffis monotonic by segments on E E EEE. For n = − 1 n = − 1 n=-1n=-1n=−1we have functions that change sign at most a finite number of times.
We will say that decomposition (1) is a decomposition of E E EEEfor the function f f fff, of order n n nnnby segments.
Any function defined on a finite set is, obviously, of order n n nnnby segments, whatever n n nnn.
If f f fffis in order n n nnnby segments on E E EEE, the function c f c f cfcfcf, Or c c cccis a constant, and, in particular, the function - f f fff, is also of order n n nnnby segments on E E EEE. This is also the case for any subset of E E EEE.
If f f fffis in order n n nnnby segments on E E EEE, any function f + P f + P f+Pf+Pf+P, Or P P PPPis a polynomial of degree n n nnn, is also of order n n nnnby segments on E E EEEWe will specify this property later.
If we can decompose the whole E E EEEinto a finite number of consecutive subsets, such that on each of these subsets the functions are of order n n nnnby segments, this function is of order n n nnnby segments on E E EEE.
2. We will say that a decomposition (I) of E E EEEfor the function f f fff, of order n n nnnby segments, is a proper decomposition if the function is not of order n n nnnon any of the sets E i + E i + 1 , i = 1 , 2 , … , m − 1 E i + E i + 1 , i = 1 , 2 , … , m − 1 E_(i)+E_(i+1),i=1,2,dots,m-1E_{i}+E_{i+1}, i=1,2, \ldots, m-1Ei+Ei+1,i=1,2,…,m−1. Otherwise, we will say that (I) is an improper decomposition. There then exists at least one E i + E i + 1 E i + E i + 1 E_(i)+E_(i+1)E_{i}+E_{i+1}Ei+Ei+1on which the function is of order n n nnn. By grouping the sets properly E i E i E_(i)E_{i}Eiwe can reduce an improper decomposition to a proper decomposition. This reduction can be carried out by successively bringing together two consecutive sets E i , E i + 1 E i , E i + 1 E_(i),E_(i+1)E_{i}, E_{i+1}Ei,Ei+1such as on E i + E i + 1 E i + E i + 1 E_(i)+E_(i+1)E_{i}+E_{i+1}Ei+Ei+1the function is of order n n nnn.
In general for an order function n n nnnby segments there are several, and even an infinity (if E E EEEis infinite), of decompositions (I) of E E EEE. The number m m mmmof subsets (I) can vary even from one proper decomposition to another proper decomposition if n ≧ 0 n ≧ 0 n >= 0n \geqq 0n≧0.
The number m m mmmsubsets of the decomposition (I) for an order function n n nnnby segments obviously has a minimum. We will designate this minimum by h h hhhand we will introduce 1a
Definition 2. The number h, minimum of m m mmm, will be called the characteristic number or simply the characteristic of the function f f fff, of order n n nnnby segments on E E EEE.
There exists, obviously, at least one decomposition (I) having exactly h h hhhterms. It is clear that any decomposition (I) having h h hhhterms is a proper decomposition. We can note, in passing, that if the characteristic of f f fffEast h h hhh, the whole E E EEEmust have at least ( n + 2 ) h − n − 1 ( n + 2 ) h − n − 1 (n+2)hn-1(n+2) hn-1(n+2)h−n−1points.
3. Let (I) be a proper decomposition of E E EEEfor the function t t ttt, of order n n nnnby segments on E E EEE. Either E 1 ∗ E 1 ∗ E_(1)^(**)E_{1}^{*}E1∗the subset of E E EEEformed by the points x ∈ E x ∈ E x in Ex \in Ex∈Esuch as f f fffeither of order n n nnnon the intersection of E E EEEwith the interval ( has , x has , x a,xa, xhas,x), has has hashashasbeing the left end of E E EEE. We see that E 1 ∗ E 1 ∗ E_(1)^(**)E_{1}^{*}E1∗is not empty since E 1 ⊆ E 1 ∗ E 1 ⊆ E 1 ∗ E_(1)subeE_(1)^(**)E_{1} \subseteq E_{1}^{*}E1⊆E1∗. We also have E 1 ∗ ⊂ E 1 + E 2 E 1 ∗ ⊂ E 1 + E 2 E_(1)^(**)subE_(1)+E_(2)E_{1}^{*} \subset E_{1}+E_{2}E1∗⊂E1+E2, since the function is not of order n n nnnon E 1 + E 2 E 1 + E 2 E_(1)+E_(2)E_{1}+E_{2}E1+E2, the decomposition (I) being, by hypothesis, a proper decomposition. The set F 2 = ( E 1 + E 2 ) − E 1 ∗ F 2 = E 1 + E 2 − E 1 ∗ F_(2)=(E_(1)+E_(2))-E_(1)^(**)F_{2}=\left(E_{1}+E_{2}\right)-E_{1}^{*}F2=(E1+E2)−E1∗is not empty and we have F 2 ⊆ E 2 F 2 ⊆ E 2 F_(2)subeE_(2)F_{2} \subseteq E_{2}F2⊆E2
The function f f fffis in order n n nnnby segments on E − E 1 ∗ E − E 1 ∗ E-E_(1)^(**)E-E_{1}^{*}E−E1∗and we have decomposition
(2) F 2 , E 3 , E 4 , … , E m (2) F 2 , E 3 , E 4 , … , E m {:(2)F_(2)","E_(3)","E_(4)","dots","E_(m):}\begin{equation*} F_{2}, E_{3}, E_{4}, \ldots, E_{m} \tag{2} \end{equation*}(2)F2,E3,E4,…,Em
of this set for the function f f fff. Decomposition (2) is, moreover, either a proper decomposition of E − E 1 ∗ E − E 1 ∗ E-E_(1)^(**)E-E_{1}^{*}E−E1∗, or else
F 2 + E 3 , E 4 , … , E m , F 2 + E 3 , E 4 , … , E m , F_(2)+E_(3),E_(4),dots,E_(m),F_{2}+E_{3}, E_{4}, \ldots, E_{m},F2+E3,E4,…,Em,
is a proper decomposition of this set.
Just as we have deduced E 1 ∗ E 1 ∗ E_(1)^(**)E_{1}^{*}E1∗of E E EEE, we can deduce E 2 ∗ E 2 ∗ E_(2)^(**)\mathrm{E}_{2}^{*}E2∗of E − E 1 ∗ E − E 1 ∗ E-E_(1)^(**)E-E_{1}^{*}E−E1∗, Then E 3 ∗ E 3 ∗ E_(3)^(**)E_{3}^{*}E3∗of E − ( E 1 ∗ + E 2 ∗ ) , … E − E 1 ∗ + E 2 ∗ , … E-(E_(1)^(**)+E_(2)^(**)),dotsE-\left(E_{1}^{*}+E_{2}^{*}\right), \ldotsE−(E1∗+E2∗),…, etc.
We immediately see that we have
Lemma I. The following E 1 ∗ , E 2 ∗ , … E 1 ∗ , E 2 ∗ , … E_(1)^(**),E_(2)^(**),dotsE_{1}^{*}, E_{2}^{*}, \ldotsE1∗,E2∗,…, is necessarily finite and has at most m m mmmterms.
From this lemma we deduce
Theorem I. The following E 1 ∗ , E 2 ∗ … E 1 ∗ , E 2 ∗ … E_(1)^(**),E_(2)^(**)dotsE_{1}^{*}, E_{2}^{*} \ldotsE1∗,E2∗…exactly h h hhhterms, h h hhhbeing the characteristic of the function f f fff.
It is clear, in fact, that this sequence is obtained independently of any proper decomposition, (I) (and even independently of any proper or improper decomposition). But there exists a proper decomposition having h h hhhterms, so the sequence has h ′ ≦ h h ′ ≦ h h^(') <= hh^{\prime} \leqq hh′≦hterms. It is clear, on the other hand, that this sequence is a proper decomposition of E E EEEfor the function f f fffDon', h ′ ≥ h h ′ ≥ h h^(') >= hh^{\prime} \geq hh′≥h, according to the definition of the number h h hhh. We have thus shown the theorem.
Definition 3. We will say that
(3) E 1 ∗ , E 2 ∗ … , E h ∗ (3) E 1 ∗ , E 2 ∗ … , E h ∗ {:(3)E_(1)^(**)","E_(2)^(**)dots","E_(h)^(**):}\begin{equation*} E_{1}^{*}, E_{2}^{*} \ldots, E_{h}^{*} \tag{3} \end{equation*}(3)E1∗,E2∗…,Eh∗
is the canonical decomposition of E E EEEfor the function f f fffof order n n nnnby segments on E E EEE.
Of course, there may be other proper decompositions having h h hhhterms.
Note that E i ∗ , E i + 1 ∗ , … , E I ∗ E i ∗ , E i + 1 ∗ , … , E I ∗ E_(i)^(**),E_(i+1)^(**),dots,E_(j)^(**)E_{i}^{*}, E_{i+1}^{*}, \ldots, E_{j}^{*}Ei∗,Ei+1∗,…,EI∗is the canonical decomposition of the sum E i ∗ + E i + 1 ∗ + … + E j ∗ , i ≧ I , j ≦ h E i ∗ + E i + 1 ∗ + … + E j ∗ , i ≧ I , j ≦ h E_(i)^(**)+E_(i+1)^(**)+dots+E_(j)^(**),i >= I,j <= hE_{i}^{*}+E_{i+1}^{*}+\ldots+E_{j}^{*}, i \geqq I, j \leqq hEi∗+Ei+1∗+…+EI∗,i≧I,I≦hand the function f f fffhas the characteristic number j − i + I j − i + I j-i+Ij-i+II−i+Ion this set.
4. Let us demonstrate the following property
Theorem 2. If h is the characteristic of the function f f fff, of order n n nnnby segments on E E EEE, the number m m mmmterms of a proper decomposition of E E EEEfor the function f f fff, is always between h h hhhAnd 2 h − I , h ≦ m ≦ 2 h − I 2 h − I , h ≦ m ≦ 2 h − I 2h-I,h <= m <= 2h-I2 h-\mathrm{I}, h \leqq m \leqq 2 h-\mathrm{I}2h−I,h≦m≦2h−I.
It is enough to demonstrate the inequality m ≤ 2 h − 1 m ≤ 2 h − 1 m <= 2h-1m \leq 2 h-1m≤2h−1. We will demonstrate the property by induction on the number h h hhh. Let (I) be a proper decomposition and (3) the canonical decomposition.
The property is true for h = 1 h = 1 h=1h=1h=1. In this case, in fact, the function is of order n n nnnon E E EEE, so we can only have m = 1 m = 1 m=1m=1m=1. Now suppose that the property is true for h − I ( h > I ) h − I ( h > I ) h-I(h > I)h-\mathrm{I}(h>\mathrm{I})h−I(h>I)and let's demonstrate it for h h hhh. Consider the sets F = E − E m , F ∗ = E − E h ∗ F = E − E m , F ∗ = E − E h ∗ F=E-E_(m),F^(**)=E-E_(h)^(**)F=E-E_{m}, F^{*}=E-E_{h}^{*}F=E−Em,F∗=E−Eh∗. The canonical decomposition of F ∗ F ∗ F^(**)F^{*}F∗East E 1 ∗ , E 2 ∗ , … , E h − 1 ∗ E 1 ∗ , E 2 ∗ , … , E h − 1 ∗ E_(1)^(**),E_(2)^(**),dots,E_(h-1)^(**)E_{1}^{*}, E_{2}^{*}, \ldots, E_{h-1}^{*}E1∗,E2∗,…,Eh−1∗. Three cases may arise:
I ∘ F ⊂ F ∗ , alors I ∘ F ⊂ F ∗ ,  alors  I^(@)F subF^(**)," alors "I^{\circ} F \subset F^{*}, \text { alors }I∘F⊂F∗, SO 
(4)
E 1 , E 2 , … , E m − 1 , F ∗ − F E 1 , E 2 , … , E m − 1 , F ∗ − F E_(1),E_(2),dots,E_(m-1),F^(**)-FE_{1}, E_{2}, \ldots, E_{m-1}, F^{*}-FE1,E2,…,Em−1,F∗−F
is a decomposition of F ∗ F ∗ F^(**)F^{*}F∗for the function f f fff. If (4) is a proper decomposition we have m ≦ 2 h − 3 m ≦ 2 h − 3 m <= 2h-3m \leqq 2 h-3m≦2h−3, therefore a fortiori m ≦ 2 h − m ≦ 2 h − m <= 2h-m \leqq 2 h-m≦2h−I. If (4) is an improper decomposition, the decomposition
(5) E 1 , E 2 , … , E m − 2 , E m − 1 + ( F ∗ − F ) (5) E 1 , E 2 , … , E m − 2 , E m − 1 + F ∗ − F {:(5)E_(1)","E_(2)","dots","E_(m-2)","E_(m-1)+(F^(**)-F):}\begin{equation*} E_{1}, E_{2}, \ldots, E_{m-2}, E_{m-1}+\left(F^{*}-F\right) \tag{5} \end{equation*}(5)E1,E2,…,Em−2,Em−1+(F∗−F)
is a proper decomposition, so m − I ≦ 2 h − 3 m − I ≦ 2 h − 3 m-I <= 2h-3m-\mathrm{I} \leqq 2 h-3m−I≦2h−3and we still have m ≦ 2 h m ≦ 2 h m <= 2hm \leqq 2 hm≦2h- I (Besides, if m = 2 m = 2 m=2m=2m=2, which can only happen if h = 2 h = 2 h=2h=2h=2, the sequence (5) is reduced to the single term F ∗ F ∗ F^(**)F^{*}F∗).
2 ∘ F = F ∗ 2 ∘ F = F ∗ 2^(@)F=F^(**)2^{\circ} F=F^{*}2∘F=F∗, SO
E 1 , E 2 , … , E m − 1 , E 1 , E 2 , … , E m − 1 , E_(1),E_(2),dots,E_(m-1),E_{1}, E_{2}, \ldots, E_{m-1},E1,E2,…,Em−1,
is a proper decomposition of F ∗ F ∗ F^(**)F^{*}F∗, SO m − I ≦ 2 h − 3 m − I ≦ 2 h − 3 m-I <= 2h-3m-\mathrm{I} \leqq 2 h-3m−I≦2h−3And m ≦ 2 h − I m ≦ 2 h − I m <= 2h-Im \leqq 2 h-\mathrm{I}m≦2h−I. 3 ∘ F ∗ ⊂ F 3 ∘ F ∗ ⊂ F 3^(@)F^(**)sub F3^{\circ} F^{*} \subset F3∘F∗⊂F, then we cannot have E m − 1 ⊆ E h ∗ E m − 1 ⊆ E h ∗ E_(m-1)subeE_(h)^(**)E_{m-1} \subseteq E_{h}^{*}Em−1⊆Eh∗, because we would have E m − 1 + E m ⊆ E h ∗ E m − 1 + E m ⊆ E h ∗ E_(m-1)+E_(m)subeE_(h)^(**)E_{m-1}+E_{m} \subseteq E_{h}^{*}Em−1+Em⊆Eh∗, contrary to the hypothesis that (I) is a proper decomposition. We see immediately that one of the sequences
E 1 , E 2 , … , E m − 2 , F ∗ − ∑ i = 1 m − 2 E i E 1 , E 2 , … , E m − 3 , E m − 2 + ( F ∗ − ∑ i = 1 m − 2 E i ) E 1 , E 2 , … , E m − 2 , F ∗ − ∑ i = 1 m − 2   E i E 1 , E 2 , … , E m − 3 , E m − 2 + F ∗ − ∑ i = 1 m − 2   E i {:[E_(1)","E_(2)","dots","E_(m-2)","F^(**)-sum_(i=1)^(m-2)E_(i)],[E_(1)","E_(2)","dots","E_(m-3)","E_(m-2)+(F^(**)-sum_(i=1)^(m-2)E_(i))]:}\begin{gathered} E_{1}, E_{2}, \ldots, E_{m-2}, F^{*}-\sum_{i=1}^{m-2} E_{i} \\ E_{1}, E_{2}, \ldots, E_{m-3}, E_{m-2}+\left(F^{*}-\sum_{i=1}^{m-2} E_{i}\right) \end{gathered}E1,E2,…,Em−2,F∗−∑i=1m−2EiE1,E2,…,Em−3,Em−2+(F∗−∑i=1m−2Ei)
is a proper decomposition of F ∗ F ∗ F^(**)F^{*}F∗. So we have m − I ≦ 2 h − 3 m − I ≦ 2 h − 3 m-I <= 2h-3m-\mathrm{I} \leqq 2 h-3m−I≦2h−3Or m − 2 ≦ 2 h − 3 m − 2 ≦ 2 h − 3 m-2 <= 2h-3m-2 \leqq 2 h-3m−2≦2h−3, so always m ≦ 2 h − I m ≦ 2 h − I m <= 2h-Im \leqq 2 h-\mathrm{I}m≦2h−I(if h = 2 h = 2 h=2h=2h=2we can easily see that m ≦ 3 m ≦ 3 m <= 3m \leqq 3m≦3).
The property is therefore also true for h h hhhand Theorem 2 follows.
The number m m mmmcan actually take all values ​​between h h hhhAnd 2 h − 1 2 h − 1 2h-12 h-12h−1. Let, in fact, be the function
f ( 3 i ) = 0 i = 1 , 2 , … , h − 1 f ( 3 i + 1 ) = 1 , i = 0 , 1 , … , h − 1 f ( 3 i + 2 ) = 2 , i = 0 , 1 , … , h − 2 f ( 3 i ) = 0      i = 1 , 2 , … , h − 1 f ( 3 i + 1 ) = 1 ,      i = 0 , 1 , … , h − 1 f ( 3 i + 2 ) = 2 ,      i = 0 , 1 , … , h − 2 {:[f(3i)=0,i=1","2","dots","h-1],[f(3i+1)=1",",i=0","1","dots","h-1],[f(3i+2)=2",",i=0","1","dots","h-2]:}\begin{array}{ll} f(3 i)=0 & i=1,2, \ldots, h-1 \\ f(3 i+1)=1, & i=0,1, \ldots, h-1 \\ f(3 i+2)=2, & i=0,1, \ldots, h-2 \end{array}f(3i)=0i=1,2,…,h−1f(3i+1)=1,i=0,1,…,h−1f(3i+2)=2,i=0,1,…,h−2
defined on the set E = { 1 , 2 , … , 3 h − 2 } E = { 1 , 2 , … , 3 h − 2 } E={1,2,dots,3h-2}E=\{1,2, \ldots, 3 h-2\}E={1,2,…,3h−2}and monotone by segments on this set. The decompositions
(6) { 1 , 2 } , { 3 , 4 , 5 } , { 6 , 7 , 8 } , … , { 3 h − 6 , 3 h − 5 , 3 h − 4 } , { 3 h − 3 , 3 h − 2 } { 1 , 2 } , { 3 , 4 , 5 } , { 6 , 7 , 8 } , … , { 3 h − 6 , 3 h − 5 , 3 h − 4 } , { 3 h − 3 , 3 h − 2 } {1,2},{3,4,5},{6,7,8},dots,{3h-6,3h-5,3h-4},{3h-3,3h-2}\{1,2\},\{3,4,5\},\{6,7,8\}, \ldots,\{3 h-6,3 h-5,3 h-4\},\{3 h-3,3 h-2\}{1,2},{3,4,5},{6,7,8},…,{3h−6,3h−5,3h−4},{3h−3,3h−2}(6) { I } , { 2 , 3 } , { 4 } , { 5 , 6 } , … , { 3 i − 2 } , { 3 i − I , 3 i } , { 3 i + I , 3 i + 2 } { I } , { 2 , 3 } , { 4 } , { 5 , 6 } , … , { 3 i − 2 } , { 3 i − I , 3 i } , { 3 i + I , 3 i + 2 } {I},{2,3},{4},{5,6},dots,{3i-2},{3i-I,3i},{3i+I,3i+2}\{\mathrm{I}\},\{2,3\},\{4\},\{5,6\}, \ldots,\{3 i-2\},\{3 i-\mathrm{I}, 3 i\},\{3 i+\mathrm{I}, 3 i+2\}{I},{2,3},{4},{5,6},…,{3i−2},{3i−I,3i},{3i+I,3i+2}, { 3 i + 3 , 3 i + 4 , 3 i + 5 } , { 3 i + 6 , 3 i + 7 , 3 i + 8 } , … { 3 i + 3 , 3 i + 4 , 3 i + 5 } , { 3 i + 6 , 3 i + 7 , 3 i + 8 } , … {3i+3,3i+4,3i+5},{3i+6,3i+7,3i+8},dots\{3 i+3,3 i+4,3 i+5\},\{3 i+6,3 i+7,3 i+8\}, \ldots{3i+3,3i+4,3i+5},{3i+6,3i+7,3i+8},…,
{ 3 h − 6 , 3 h − 5 , 3 h − 4 } , { 3 h − 3 , 3 h − 2 } , { 3 h − 6 , 3 h − 5 , 3 h − 4 } , { 3 h − 3 , 3 h − 2 } , {3h-6,3h-5,3h-4},{3h-3,3h-2},\{3 h-6,3 h-5,3 h-4\},\{3 h-3,3 h-2\},{3h−6,3h−5,3h−4},{3h−3,3h−2},
( 6 h − 2 ) { I } , { 2 , 3 } , { 4 } { 5 , 6 } , … , { 3 h − 8 } , { 3 h − 7 , 3 h − 6 } , { 3 h − 5 , 3 h − 4 } 6 h − 2 { I } , { 2 , 3 } , { 4 } { 5 , 6 } , … , { 3 h − 8 } , { 3 h − 7 , 3 h − 6 } , { 3 h − 5 , 3 h − 4 } (6_(h-2)){I},{2,3},{4}{5,6},dots,{3h-8},{3h-7,3h-6},{3h-5,3h-4}\left(6_{h-2}\right)\{I\},\{2,3\},\{4\}\{5,6\}, \ldots,\{3 h-8\},\{3 h-7,3 h-6\},\{3 h-5,3 h-4\}(6h−2){I},{2,3},{4}{5,6},…,{3h−8},{3h−7,3h−6},{3h−5,3h−4}
{ 3 h − 3 , 3 h − 2 } , { 3 h − 3 , 3 h − 2 } , {3h-3,3h-2},\{3 h-3,3 h-2\},{3h−3,3h−2},
( 6 h − 1 ) { 1 } , { 2 , 3 } , { 4 } , { 5 , 6 } , … , { 3 h − 5 } , { 3 h − 4 , 3 h − 3 } , { 3 h − 2 } 6 h − 1 { 1 } , { 2 , 3 } , { 4 } , { 5 , 6 } , … , { 3 h − 5 } , { 3 h − 4 , 3 h − 3 } , { 3 h − 2 } (6_(h-1)){1},{2,3},{4},{5,6},dots,{3h-5},{3h-4,3h-3},{3h-2}\left(6_{h-1}\right)\{1\},\{2,3\},\{4\},\{5,6\}, \ldots,\{3 h-5\},\{3 h-4,3 h-3\},\{3 h-2\}(6h−1){1},{2,3},{4},{5,6},…,{3h−5},{3h−4,3h−3},{3h−2}, are proper decompositions of E E EEE. The decomposition (6i) has h + i h + i h+ih+ih+iterms, i = 0 , 1 , … , h − 1 i = 0 , 1 , … , h − 1 i=0,1,dots,h-1i=0,1, \ldots, h-1i=0,1,…,h−1. The decomposition ( 6 0 6 0 6_(0)6_{0}60) is precisely the canonical decomposition.
The number h h hhhis always between m + I 2 m + I 2 (m+I)/(2)\frac{m+I}{2}m+I2And m m mmm. It follows that if m = 2 m = 2 m=2m=2m=2, we must have h = 2 h = 2 h=2h=2h=2. If m > 2 , h m > 2 , h m > 2,hm>2, hm>2,hmay be smaller than m m mmmunless n = − I n = − I n=-In=-\mathrm{I}n=−I, when we have the
Theorem 3. If n = − 1 n = − 1 n=-1n=-1n=−1, any proper decomposition of E E EEEhas h h hhhterms. Indeed, if (I) is a proper decomposition of E E EEE, we can find the points x i ∈ E i , i = 1 , 2 , … , m x i ∈ E i , i = 1 , 2 , … , m x_(i)inE_(i),i=1,2,dots,mx_{i} \in E_{i}, i=1,2, \ldots, mxi∈Ei,i=1,2,…,m, such as f ( x i ) f ( x i + 1 ) < 0 , i = 1 , 2 f x i f x i + 1 < 0 , i = 1 , 2 f(x_(i))f(x_(i+1)) < 0,i=1,2f\left(x_{i}\right) f\left(x_{i+1}\right)<0, i=1,2f(xi)f(xi+1)<0,i=1,2, ⋯ , m − I ⋯ , m − I cdots,m-I\cdots, m-I⋯,m−I. Two consecutive points x i , x i + 1 x i , x i + 1 x_(i),x_(i+1)x_{i}, x_{i+1}xi,xi+1cannot therefore belong to the same E j ∗ E j ∗ E_(j)^(**)E_{j}^{*}EI∗. So we have m ≦ h m ≦ h m <= hm \leqq hm≦h. On the other hand, m ≥ h m ≥ h m >= hm \geq hm≥h, according to the challenge of the number h h hhh. Done m = h m = h m=hm=hm=h.
Cernăuți, April 22, 1940.
1940

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