On the extension of Lipschitz functions

Costica Mustata
(original), Approximation Theory, paper

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Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

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C. Mustăţa, On the extension of Lipschitz functions, Seminar of Funct. Analysis and Numerical Methods, Preprint Nr. 1 (1986), 83-92.

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[1] Cobzas, S., Mustata, C., Norm Preserving Exctension of Convex Lipschitz Functions, J.A.T., 24 (1978), 236-244.
[2] Dunham, C.B., Chebyshev approximation with a null space, Proc. Amer. Math. Soc. 41 (1973) 557-558.
[3] Johnson, J.A., Banach Spaces of Lipschitz Functions and Vector-Valued Lipschitz Funcitons, Trans. Amer. Math. Soc. 148 (1970), 147-169.
[4] Shane, E.J., Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842.
[5] Mustata, C., Best Approximaiton and Unique Extension of Lipschitz Functions, J.A.T., 19 (1977), 222-230.
[6] Roy, A.K., Extreme Points and Linear Isometries of Banach Space of Lipschitz Functions, Canad. J. of Math. 20 (1968), 1150-1164.
[7] Singer, I., Cea mai bună aproximare în spații vectoriale normate prin elemente din subspații vectoriale, Edit. Acad. R.S. Română, București, 1967.

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1986-Mustata-UBB-Seminar-On-the-extension-of-Lipschitz-functions

ON THE EXTENSION OF LIPSCHITZ FUNCTIONS

Costică Mustăta

  1. Introduction. Let A A AAA be a subset of the interval [ a , b ] ⊂ R [ a , b ] ⊂ R [a,b]sub R[a, b] \subset R[a,b]⊂R. A function f : A → R f : A → R f:A rarr Rf: A \rightarrow Rf:A→R is called Lipschitz if there exists L ⩾ 0 L ⩾ 0 L >= 0L \geqslant 0L⩾0 such that
(1) | f ( x ) − f ( y ) | ⩽ L ⋅ | x − y | , (1) | f ( x ) − f ( y ) | ⩽ L ⋅ | x − y | , {:(1)|f(x)-f(y)| <= L*|x-y|",":}\begin{equation*} |f(x)-f(y)| \leqslant L \cdot|x-y|, \tag{1} \end{equation*}(1)|f(x)−f(y)|⩽L⋅|x−y|,
for all x , y ∈ A x , y ∈ A x,y in Ax, y \in Ax,y∈A. The smallest number L L LLL for which the inerguality
(1) holds is called the Lipschitz norm of f f fff and is denotod by ‖ f ‖ L ‖ f ‖ L ||f||_(L)\|\mathbf{f}\|_{L}‖f‖L. The Lipschitz norm of f f fff can be calculated also by the formula
(2) ‖ f ‖ L = sup { | f ( x ) − f ( y ) | / | x − y | : x , y ∈ A , x ≠ y } ‖ f ‖ L = sup { | f ( x ) − f ( y ) | / | x − y | : x , y ∈ A , x ≠ y } ||f||_(L)=s u p{|f(x)-f(y)|//|x-y|:x,y in A,x!=y}\|f\|_{L}=\sup \{|f(x)-f(y)| /|x-y|: x, y \in A, x \neq y\}‖f‖L=sup{|f(x)−f(y)|/|x−y|:x,y∈A,x≠y}.
Denote by Lip A the set of all real valued Lipschitz func tions on A, i.e.
(3) Lip A = { P , f : A → R , f Lip A = { P , f : A → R , f quad Lip A={P,f:A rarr R,f\quad \operatorname{Lip} A=\{P, f: A \rightarrow R, fLipA={P,f:A→R,f is Lipschitz } } }\}}.
With the usual (i.e. pointwise ) operations of addition and multiplication by scalars, Lip A is a vector space.
A Lipschitz extension of f f fff to [ a , b ] [ a , b ] [a,b][a, b][a,b] is a Lipschitz function [ a , b ] → R [ a , b ] → R [a,b]rarrR[\mathrm{a}, \mathrm{b}] \rightarrow \mathrm{R}[a,b]→R such that
F | A ′ = P and ‖ F ‖ L = ‖ I ‖ L F A ′ = P  and  ‖ F ‖ L = ‖ I ‖ L F|_(A)^(')=P" and "||F||_(L)=||I||_(L)\left.F\right|_{A} ^{\prime}=P \text { and }\|F\|_{L}=\|I\|_{L}F|A′=P and ‖F‖L=‖I‖L
By a result of Mo SHANE [4] every function in Iip A A AAA has at least one Lipschitz extension in Lip [ a , b ] Lip [ a , b ] Lip[a,b]\operatorname{Lip}[\mathrm{a}, \mathrm{b}]Lip[a,b]. More exactly, the following two functions
F 1 ( x ) = sup { f ( y ) − ‖ f ‖ L ⋅ | x − y | : y ∈ A } F 1 ( x ) = sup f ( y ) − ‖ f ‖ L ⋅ | x − y | : y ∈ A F_(1)(x)=s u p{f(y)-||f||_(L)*|x-y|:y in A}F_{1}(x)=\sup \left\{f(y)-\|f\|_{L} \cdot|x-y|: y \in A\right\}F1(x)=sup{f(y)−‖f‖L⋅|x−y|:y∈A}
and.
(6) F 2 ( x ) = inf { f ( y ) + ‖ f ‖ I ′ ⋅ | x − y | : y ∈ A } (6) F 2 ( x ) = inf f ( y ) + ‖ f ‖ I ′ ⋅ | x − y | : y ∈ A {:(6)F_(2)(x)=i n f{f(y)+||f||_(I^('))*|x-y|quad:quad y inA}:}\begin{equation*} F_{2}(x)=\inf \left\{f(y)+\|f\|_{I^{\prime}} \cdot|x-y| \quad: \quad y \in \mathbb{A}\right\} \tag{6} \end{equation*}(6)F2(x)=inf{f(y)+‖f‖I′⋅|x−y|:y∈A}
are Lipschitz extensions of f f fff to [ a , b ] [ a , b ] [a,b][a, b][a,b]. Denoting by E ( f ; [ a , b ] ) E ( f ; [ a , b ] ) E(f;[a,b])E(f ;[a, b])E(f;[a,b]) the set of all Lipschitz extensions of f f fff to [ a , b ] [ a , b ] [a,b][a, b][a,b], i.e.
(7) E ( f ; [ a , b ] ) = { F ∈ Lip [ a , b ] : F | A = f E ( f ; [ a , b ] ) = F ∈ Lip [ a , b ] : F A = f E(f;[a,b])={F in Lip[a,b]:F|_(A)=f:}\mathbb{E}(f ;[a, b])=\left\{F \in \operatorname{Lip}[a, b]:\left.F\right|_{A}=f\right.E(f;[a,b])={F∈Lip[a,b]:F|A=f and ‖ F ‖ L = ‖ f ‖ L } ‖ F ‖ L = ‖ f ‖ L {:||F||_(L)=||f||_(L)}\left.\|F\|_{L}=\|f\|_{L}\right\}‖F‖L=‖f‖L}
the following assertions hold true
(a) F 1 ( x ) ≤ F ( x ) ≤ F 2 ( x ) , x ∈ [ a , b ] F 1 ( x ) ≤ F ( x ) ≤ F 2 ( x ) , x ∈ [ a , b ] quadF_(1)(x) <= F(x) <= F_(2)(x),x in[a,b]\quad F_{1}(x) \leq F(x) \leq F_{2}(x), x \in[a, b]F1(x)≤F(x)≤F2(x),x∈[a,b], for all F ∈ R ( f ; [ a , b ] ) F ∈ R ( f ; [ a , b ] ) F inR(f;[a,b])F \in \mathbb{R}(f ;[a, b])F∈R(f;[a,b]);
(b) E ( f ; [ a , b ] ) E ( f ; [ a , b ] ) quad E(f;[a,b])\quad E(f ;[a, b])E(f;[a,b]) is a convex subset of Lip [ a , b ] Lip [ a , b ] Lip[a,b]\operatorname{Lip}[a, b]Lip[a,b];
(c) The functions F 1 F 1 F_(1)\mathrm{F}_{1}F1 and F 2 F 2 F_(2)\mathrm{F}_{2}F2 are extreme points of E ( f ; [ a , b ] ) E ( f ; [ a , b ] ) E(f;[a,b])E(f ;[a, b])E(f;[a,b]).
By the definition of the Lipschitz noxm ( or by (2) ), ‖ f ‖ L = 0 ‖ f ‖ L = 0 ||f||_(L)=0\|f\|_{L}=0‖f‖L=0 if and only if f = f = f=f=f= constant and therefore ‖ ‖ ||\|‖." I I III is not actually a norm on Lip A Lip A Lip A\operatorname{Lip} ALipA but it is a norm on the space Lip 0 0 _(0){ }_{0}0 of all functions in Lip A A AAA vanishing at a fixed point x 0 ∈ A x 0 ∈ A x_(0)in Ax_{0} \in Ax0∈A. The space Lip A A A A _(A)^(A){ }_{\mathrm{A}}{ }^{\mathrm{A}}AA with the Lipschitz norm is a dual Banach space (see [3]).
2. Lipschitz extensions from finite subsets of [ a , b ] [ a , b ] [a,b][a, b][a,b].
Let C [ a , b ] C [ a , b ] C[a,b]C[a, b]C[a,b] be the space of all real valued continuous functions on [ a , b ] [ a , b ] [a,b][a, b][a,b] and let
(8) u = { x 0 , x 1 , … , x n } , a ⩽ x 0 < x 1 < … < x n ⩽ b , (8) u = x 0 , x 1 , … , x n , a ⩽ x 0 < x 1 < … < x n ⩽ b , {:(8)u={x_(0),x_(1),dots,x_(n)}quad","quad a <= x_(0) < x_(1) < dots < x_(n) <= b quad",":}\begin{equation*} u=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\} \quad, \quad a \leqslant x_{0}<x_{1}<\ldots<x_{n} \leqslant b \quad, \tag{8} \end{equation*}(8)u={x0,x1,…,xn},a⩽x0<x1<…<xn⩽b,
be a linite subset of [ a , b ] [ a , b ] [a,b][\mathrm{a}, \mathrm{b}][a,b]. Then obviously, the restriction f | 11 f 11 f|_(11)\left.f\right|_{11}f|11 of a a aaa function f ∈ C [ a , b ] f ∈ C [ a , b ] f in C[a,b]f \in C[a, b]f∈C[a,b] to M M MMM is in Lip M Lip M Lip M\operatorname{Lip} MLipM and
(9) | f | M ‖ L = max { | f ( x i ) − f ( x j ) | / | x i − x j | : i , j = 0 , 1 , … n , i ≠ j } | f | M ‖ L = max f x i − f x j / x i − x j : i , j = 0 , 1 , … n , i ≠ j quad|f|_(M)||_(L)=max{|f(x_(i))-f(x_(j))|//|x_(i)-x_(j)|:i,j=0,1,dots n,i!=j}\quad|f|_{M} \|_{L}=\max \left\{\left|f\left(x_{i}\right)-f\left(x_{j}\right)\right| /\left|x_{i}-x_{j}\right|: i, j=0,1, \ldots n, i \neq j\right\}|f|M‖L=max{|f(xi)−f(xj)|/|xi−xj|:i,j=0,1,…n,i≠j} Let U : C [ a , b ] → Lip M U : C [ a , b ] → Lip M U:C[a,b]rarr Lip MU: C[a, b] \rightarrow \operatorname{Lip} MU:C[a,b]→LipM be the restriction operator, i.e.
(10)
U ( f ) = f | M , for f ∈ C [ a , b ] U ( f ) = f M ,  for  f ∈ C [ a , b ] U(f)=f|_(M)," for "f in C[a,b]U(f)=\left.f\right|_{M}, \text { for } f \in C[a, b]U(f)=f|M, for f∈C[a,b]
By the above quated result of HC SHANE, f | M f M f|_(M)\left.f\right|_{M}f|M has at least a Lipschitz extension F ∈ Lip [ a , b ] F ∈ Lip [ a , b ] F in Lip[a,b]F \in \operatorname{Lip}[a, b]F∈Lip[a,b]. Let V : Lip M → P ( Lip [ a , b ] ) V : Lip M → P ( Lip [ a , b ] ) V:Lip M rarrP(Lip[a,b])V: \operatorname{Lip} M \rightarrow \mathscr{P}(\operatorname{Lip}[a, b])V:LipM→P(Lip[a,b]) the extension operator defined by
(11) V ( g ) = E ( g ; [ a , b ] ) , g ∈ Lip M , (11) V ( g ) = E ( g ; [ a , b ] ) , g ∈ Lip M , {:(11)V(g)=E(g;[a","b])","quad g in Lip M",":}\begin{equation*} V(g)=E(g ;[a, b]), \quad g \in \operatorname{Lip} M, \tag{11} \end{equation*}(11)V(g)=E(g;[a,b]),g∈LipM,
and let W : C [ a , b ] → G ρ ( Lip [ a , b ] ) W : C [ a , b ] → G ρ ( Lip [ a , b ] ) W:C[a,b]rarrG^(rho)(Lip[a,b])W: C[a, b] \rightarrow \mathcal{G}^{\rho}(\operatorname{Lip}[a, b])W:C[a,b]→Gρ(Lip[a,b]) be the composition of U U UUU and V V VVV
(12) W = V 0 U (12) W = V 0 U {:(12)W=V_(0)U:}\begin{equation*} W=V_{0} U \tag{12} \end{equation*}(12)W=V0U
In general W W WWW is a multivalued operator (point to set) . A function f ∈ C [ a , b ] f ∈ C [ a , b ] f in C[a,b]f \in C[a, b]f∈C[a,b] such that f ∈ W ( f ) f ∈ W ( f ) f in W(f)f \in W(f)f∈W(f) is called a fix point of W W WWW.
Obviously, the set of fix points of the operator W W WWW is non-void.
Indeed, if g ∈ Lip M g ∈ Lip M g in Lip Mg \in \operatorname{Lip} Mg∈LipM and f f fff is a Lipschitz extension of G G GGG to [ a , b ] [ a , b ] [a,b][a, b][a,b], then f ∈ W ( f ) f ∈ W ( f ) f in W(f)f \in W(f)f∈W(f). The fix points of the operator W W WWW are characterized in Theorem 2.1 below.
For x , y ∈ [ a , b ] , x ≠ y and f ∈ C [ a , b ] , put  For  x , y ∈ [ a , b ] , x ≠ y  and  f ∈ C [ a , b ] , put  " For "x,y in[a,b],x!=y" and "f in C[a,b]", put "\text { For } x, y \in[a, b], x \neq y \text { and } f \in C[a, b] \text {, put } For x,y∈[a,b],x≠y and f∈C[a,b], put 
(13) [ x , y ; f ] = ( f ( x ) − f ( y ) ) / ( x − y ) (13) [ x , y ; f ] = ( f ( x ) − f ( y ) ) / ( x − y ) {:(13)[x","y;f]=(f(x)-f(y))//(x-y):}\begin{equation*} [x, y ; f]=(f(x)-f(y)) /(x-y) \tag{13} \end{equation*}(13)[x,y;f]=(f(x)−f(y))/(x−y)
and
(14) f ( x , y ; f ) ( t ) = [ x , y ; f ] ⋅ ( t − x ) + f ( x ) , t ∈ [ a , b ] (14) f ( x , y ; f ) ( t ) = [ x , y ; f ] ⋅ ( t − x ) + f ( x ) , t ∈ [ a , b ] {:(14)f(x","y;f)(t)=[x","y;f]*(t-x)+f(x)","quad t in[a","b]:}\begin{equation*} f(x, y ; f)(t)=[x, y ; f] \cdot(t-x)+f(x), \quad t \in[a, b] \tag{14} \end{equation*}(14)f(x,y;f)(t)=[x,y;f]⋅(t−x)+f(x),t∈[a,b]
2.1 THEORTM, Let f ∈ C [ a , b ] f ∈ C [ a , b ] finC[a,b]\mathrm{f} \in \mathrm{C}[\mathrm{a}, \mathrm{b}]f∈C[a,b] and let M be the set ( e ).
Then f ∈ W ( f ) f ∈ W ( f ) f in W(f)f \in W(f)f∈W(f) if and only if there exists an index k ∈ { 0 , 1 , … , n − 1 } k ∈ { 0 , 1 , … , n − 1 } k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\}k∈{0,1,…,n−1} such that
(15) sup { | [ x , y ; f ] | : x , y ∈ [ a , b ] , x ∉ y } = | [ x k , x k + 1 ; f ] | sup { | [ x , y ; f ] | : x , y ∈ [ a , b ] , x ∉ y } = x k , x k + 1 ; f s u p{|[x,y;f]|:quad x,y in[a,b],quad x!in y}=|[x_(k),x_(k+1);f]|\sup \{|[x, y ; f]|: \quad x, y \in[a, b], \quad x \notin y\}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|sup{|[x,y;f]|:x,y∈[a,b],x∉y}=|[xk,xk+1;f]|.
Proof. If f ∈ W ( f ) f ∈ W ( f ) f in W(f)f \in W(f)f∈W(f) then f ∈ E ( f | M ; [ a , b ] ) f ∈ E f M ; [ a , b ] f in E(f|_(M);[a,b])f \in E\left(\left.f\right|_{M} ;[a, b]\right)f∈E(f|M;[a,b]) and
‖ I ‖ L = ‖ f ‖ M ‖ L = max { | [ x j , x j + 1 ; f ] | : j = 0 , 1 , … , n − 1 } ‖ I ‖ L = ‖ f ‖ M ‖ L = max x j , x j + 1 ; f : j = 0 , 1 , … , n − 1 ||I||_(L)=||f||_(M)||_(L)=max{|[x_(j),x_(j+1);f]|:j=0,1,dots,n-1}\|I\|_{L}=\|f\|_{M} \|_{L}=\max \left\{\left|\left[x_{j}, x_{j+1} ; f\right]\right|: j=0,1, \ldots, n-1\right\}‖I‖L=‖f‖M‖L=max{|[xj,xj+1;f]|:j=0,1,…,n−1}
so that, there exists k ∈ { 0 , 1 , … , n − 1 } k ∈ { 0 , 1 , … , n − 1 } k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\}k∈{0,1,…,n−1} such that ‖ f ‖ M ‖ L = ‖ f ‖ M ‖ L = ||f||_(M)||_(L)=\|f\|_{M} \|_{L}=‖f‖M‖L=
| [ x k , x k + 1 ; f ] | x k , x k + 1 ; f |[x_(k),x_(k+1);f]|\left|\left[x_{k}, x_{k+1} ; f\right]\right||[xk,xk+1;f]|, and the relation (15) holds.
Conversely, if the relation (15) holds for a index k ∈ { 0 , 1 , … k ∈ { 0 , 1 , … k in{0,1,dotsk \in\{0,1, \ldotsk∈{0,1,…, n − 1 } n − 1 } n-1}n-1\}n−1}, then
| [ x k , x k + 1 ; f ] | = sup { | [ x , y ; f ] | : x , y ∈ [ a , b ] , x ≠ y } ⩾ x k , x k + 1 ; f = sup { | [ x , y ; f ] | : x , y ∈ [ a , b ] , x ≠ y } ⩾ |[x_(k),x_(k+1);f]|=s u p{|[x,y;f]|:x,y in[a,b],x!=y} >=\left|\left[x_{k}, x_{k+1} ; f\right]\right|=\sup \{|[x, y ; f]|: x, y \in[a, b], x \neq y\} \geqslant|[xk,xk+1;f]|=sup{|[x,y;f]|:x,y∈[a,b],x≠y}⩾
⩾ max { | [ x j , x j + 1 ; f ] | : j = 0 , 1 , … , n − 1 } = ‖ f | M ‖ L ⩾ max x j , x j + 1 ; f : j = 0 , 1 , … , n − 1 = f M L >= max{|[x_(j),x_(j+1);f]|:j=0,1,dots,n-1}=||f|_(M)||_(L)\geqslant \max \left\{\left|\left[x_{j}, x_{j+1} ; f\right]\right|: j=0,1, \ldots, n-1\right\}=\left\|\left.f\right|_{M}\right\|_{L}⩾max{|[xj,xj+1;f]|:j=0,1,…,n−1}=‖f|M‖L.
Therefore, | f | ‖ I = | [ x k , x k + 1 ; f ] | = ‖ f ‖ L | f | I = x k , x k + 1 ; f = f ‖ L |f|||_(I)=|[x_(k),x_(k+1);f]|=||f||_(L)|f|\left\|_{I}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|=\right\| f \|_{L}|f|‖I=|[xk,xk+1;f]|=‖f‖L, which shows that f ∈ W ( f ) f ∈ W ( f ) f in W(f)f \in W(f)f∈W(f).
Theoret 2.1 has some corollaries.
2.2 COROITARY . If the relation (15) holds for an index k ∈ { 0 , 1 , … , n − 1 } k ∈ { 0 , 1 , … , n − 1 } k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\}k∈{0,1,…,n−1} then
f ( x ) = L ( x k , x k + 1 ; f ) ( x ) for all x ∈ [ x k , x k + 1 ] f ( x ) = L x k , x k + 1 ; f ( x )  for all  x ∈ x k , x k + 1 f(x)=L(x_(k),x_(k+1);f)(x)" for all "x in[x_(k),x_(k+1)]f(x)=\mathcal{L}\left(x_{k}, x_{k+1} ; f\right)(x) \text { for all } x \in\left[x_{k}, x_{k+1}\right]f(x)=L(xk,xk+1;f)(x) for all x∈[xk,xk+1]
Proof. If f ( x ′ ) ≠ ℓ ( x k , x k + 1 ; f ) ( x ′ ) f x ′ ≠ ℓ x k , x k + 1 ; f x ′ f(x^('))!=ℓ(x_(k),x_(k+1);f)(x^('))f\left(x^{\prime}\right) \neq \ell\left(x_{k}, x_{k+1} ; f\right)\left(x^{\prime}\right)f(x′)≠ℓ(xk,xk+1;f)(x′), for an x ′ ∈ ( x k , x k + 1 ) x ′ ∈ x k , x k + 1 x^(')in(x_(k),x_(k+1))x^{\prime} \in\left(x_{k}, x_{k+1}\right)x′∈(xk,xk+1)
then
max { | [ x k , x ′ ; f ] | , | [ x ′ , x k + 1 ; f ] | } > | [ x k , x k + 1 ; f ] | = ⋅ = sup { | [ x , y ; f ] | : x , y ∈ [ a , b ] , x ≠ y } max x k , x ′ ; f , x ′ , x k + 1 ; f > x k , x k + 1 ; f = ⋅ = sup { | [ x , y ; f ] | : x , y ∈ [ a , b ] , x ≠ y } {:[max{|[x_(k),x^(');f]|,:}{:|[x^('),x_(k+1);f]|} > |[x_(k),x_(k+1);f]|=],[*=s u p{|[x","y;f]|:x","y in[a","b]","x!=y}]:}\begin{aligned} \max \left\{\left|\left[x_{k}, x^{\prime} ; f\right]\right|,\right. & \left.\left|\left[x^{\prime}, x_{k+1} ; f\right]\right|\right\}>\left|\left[x_{k}, x_{k+1} ; f\right]\right|= \\ \cdot & =\sup \{|[x, y ; f]|: x, y \in[a, b], x \neq y\} \end{aligned}max{|[xk,x′;f]|,|[x′,xk+1;f]|}>|[xk,xk+1;f]|=⋅=sup{|[x,y;f]|:x,y∈[a,b],x≠y}
which is a contradiction.
2.3 COROILARY. If f ∈ W ( f ) f ∈ W ( f ) f in W(f)f \in W(f)f∈W(f) then f ( x ) = F ( x ) , x ∈ [ x k , x k + 1 ] f ( x ) = F ( x ) , x ∈ x k , x k + 1 f(x)=F(x),x in[x_(k),x_(k+1)]f(x)=F(x), x \in\left[x_{k}, x_{k+1}\right]f(x)=F(x),x∈[xk,xk+1], for 311. F ∈ W ( f ) F ∈ W ( f ) F in W(f)F \in W(f)F∈W(f), where k ∈ { 0 , 1 , … , n − 1 } k ∈ { 0 , 1 , … , n − 1 } k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\}k∈{0,1,…,n−1} is the index for which relation (25) is true.
Proof. If F ∈ W ( ± ) F ∈ W ( ± ) F in W(+-)F \in W( \pm)F∈W(±) then
sup { | [ x , y ; F ] | : x , y ∈ [ a , b ] , x ≠ y } = ‖ F ‖ I = ‖ f ‖ M ‖ L =∣ [ x k , x k + 1 ; F ‖ F ∣ sup { | [ x , y ; F ] | : x , y ∈ [ a , b ] , x ≠ y } = ‖ F ‖ I = ‖ f ‖ M ‖ L =∣ x k , x k + 1 ; F ‖ F ∣ s u p{|[x,y;F]|:quadx,yin[a,b],x!=y}=||F||_(I)=||f||_(M)||_(L)=∣[x_(k),x_(k+1);F||^(F)∣:}\sup \{|[\mathrm{x}, \mathrm{y} ; \mathrm{F}]|: \quad \mathrm{x}, \mathrm{y} \in[\mathrm{a}, \mathrm{b}], \mathrm{x} \neq \mathrm{y}\}=\|\mathrm{F}\|_{\mathrm{I}}=\|\mathrm{f}\|_{\mathrm{M}} \|_{\mathrm{L}}=\mid\left[\mathrm{x}_{\mathrm{k}}, \mathrm{x}_{\mathrm{k}+1} ; \mathrm{F} \|^{\mathrm{F}} \mid\right.sup{|[x,y;F]|:x,y∈[a,b],x≠y}=‖F‖I=‖f‖M‖L=∣[xk,xk+1;F‖F∣
and Corollary - 2.3 follows from Corollary 2.2 .
2.4 COROTLARY. If a = x 0 a = x 0 a=x_(0)a=x_{0}a=x0 and x n = b x n = b x_(n)=bx_{n}=bxn=b, then
(a) sup { | [ x , y ; f ] | : x , y ∈ [ a , b ] , x ≠ y } = | [ x 0 , x n ; f ] | sup { | [ x , y ; f ] | : x , y ∈ [ a , b ] , x ≠ y } = x 0 , x n ; f s u p{|[x,y;f]|:x,y in[a,b],x!=y}=|[x_(0),x_(n);f]|\sup \{|[x, y ; f]|: x, y \in[a, b], x \neq y\}=\left|\left[x_{0}, x_{n} ; f\right]\right|sup{|[x,y;f]|:x,y∈[a,b],x≠y}=|[x0,xn;f]|, implies f ( x ) = f ( x 0 , x n ; f ) ( x ) f ( x ) = f x 0 , x n ; f ( x ) f(x)=f(x_(0),x_(n);f)(x)f(x)=f\left(x_{0}, x_{n} ; f\right)(x)f(x)=f(x0,xn;f)(x), for all x ∈ [ a , b ] x ∈ [ a , b ] x in[a,b]x \in[a, b]x∈[a,b], and
(b) sup { | [ x , y ; e ] | : x , y ∈ { a , b ] , x ≠ y } = | [ x k , x k + 1 ; f ] | sup { | [ x , y ; e ] | : x , y ∈ { a , b ] , x ≠ y } = x k , x k + 1 ; f s u p{|[x,y;e]|:x,y in{a,b],x!=y}=|[x_(k),x_(k+1);f]|\sup \{|[x, y ; e]|: x, y \in\{a, b], x \neq y\}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|sup{|[x,y;e]|:x,y∈{a,b],x≠y}=|[xk,xk+1;f]|, k = 0 , 1 , … , n − 1 k = 0 , 1 , … , n − 1 k=0,1,dots,n-1k=0,1, \ldots, n-1k=0,1,…,n−1, implies f ( x ) = f ( x k , x k + 1 ; f ) ( x ) f ( x ) = f x k , x k + 1 ; f ( x ) f(x)=f(x_(k),x_(k+1);f)(x)f(x)=f\left(x_{k}, x_{k+1} ; f\right)(x)f(x)=f(xk,xk+1;f)(x), x ∈ [ x k , x k + 1 ] , k = 0 , 1 , … , n − 1 x ∈ x k , x k + 1 , k = 0 , 1 , … , n − 1 x in[x_(k),x_(k+1)]quad,k=0,1,dots,n-1x \in\left[x_{k}, x_{k+1}\right] \quad, k=0,1, \ldots, n-1x∈[xk,xk+1],k=0,1,…,n−1.
Combining Corollaries 2.3 and 2.4 it follows
2.5 COROLLARY. If f ∈ Iip [ a , b ] , x o = a , x n = b f ∈ Iip [ a , b ] , x o = a , x n = b f in Iip[a,b],x_(o)=a,x_(n)=bf \in \operatorname{Iip}[a, b], x_{o}=a, x_{n}=bf∈Iip[a,b],xo=a,xn=b and ‖ f ‖ L ==∣ [ x k , x k + 1 ; f ] , k = 0 , 1 , … , n − 1 ‖ f ‖ L ==∣ x k , x k + 1 ; f , k = 0 , 1 , … , n − 1 ||f||_(L)==∣[x_(k),x_(k+1);f],k=0,1,dots,n-1\|f\|_{L}= =\mid\left[x_{k}, x_{k+1} ; f\right], k=0,1, \ldots, n-1‖f‖L==∣[xk,xk+1;f],k=0,1,…,n−1, then W ( f ) = { f } W ( f ) = { f } W(f)={f}W(f)=\{f\}W(f)={f}.
3. Faces and extreme points.
Let X X XXX be a normed space and B ( X ) = { X G X : ‖ x ‖ ≤ 1 } B ( X ) = { X G X : ‖ x ‖ ≤ 1 } B(X)={XGX:||x|| <= 1}B(X)=\{X G X:\|x\| \leq 1\}B(X)={XGX:‖x‖≤1} its closed unit ball. A subset A ⊆ B ( X ) A ⊆ B ( X ) A sube B(X)A \subseteq B(X)A⊆B(X) is called an extremal subset (a face) of B ( X ) B ( X ) B(X)B(X)B(X) if α f 1 + ( 1 − α ) f 2 ∈ A α f 1 + ( 1 − α ) f 2 ∈ A alphaf_(1)+(1-alpha)f_(2)in A\alpha f_{1}+(1-\alpha) f_{2} \in Aαf1+(1−α)f2∈A for f 1 , f 2 ∈ B ( X ) f 1 , f 2 ∈ B ( X ) f_(1),f_(2)in B(X)f_{1}, f_{2} \in B(X)f1,f2∈B(X) and a number α , 0 < α < 1 α , 0 < α < 1 alpha,0 < alpha < 1\alpha, 0<\alpha<1α,0<α<1, implies f 1 , f 2 ∈ A f 1 , f 2 ∈ A f_(1),f_(2)in Af_{1}, f_{2} \in Af1,f2∈A. If A A AAA contains exactly one point f f fff, then f f fff is called an extreme point of B ( X ) B ( X ) B(X)B(X)B(X).
Let M M MMM be a finite set of real numbers
and
M = { x 0 , x 1 , … , x n } , x 0 < x 1 < … < x n M = x 0 , x 1 , … , x n , x 0 < x 1 < … < x n M={x_(0),x_(1),dots,x_(n)}quad,quadx_(0) < x_(1) < dots < x_(n)M=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\} \quad, \quad x_{0}<x_{1}<\ldots<x_{n}M={x0,x1,…,xn},x0<x1<…<xn
Lip 0 M ′ = { f , Lip 0 M ′ = f , Lip_(0)M^(')={f,quad:}\operatorname{Lip}_{0} M^{\prime}=\left\{f, \quad\right.Lip0M′={f, f : M → R , f : M → R , f :M rarr R,quad f: M \rightarrow R, \quad f:M→R,f is Lipschitz and f ( x 0 ) = 0 } f x 0 = 0 {:f(x_(0))=0}\left.f\left(x_{0}\right)=0\right\}f(x0)=0}.
3.1 THEOREM. The function f ∈ B ( f ∈ B f in B(:}f \in B\left(\right.f∈B( Lip 0 M ) 0 M {:_(0)M)\left._{0} M\right)0M) is an extreme point of B ( Lip 0 M ) B Lip 0 M B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right)B(Lip0M) if and only if
(16) ‖ f ‖ L = 1 =∣ [ x k , x k + 1 ; f ] ] (16) ‖ f ‖ L = 1 =∣ x k , x k + 1 ; f {:(16){:||f||_(L)=1=∣[x_(k),x_(k+1);f]]:}\begin{equation*} \left.\|f\|_{L}=1=\mid\left[x_{k}, x_{k+1} ; f\right]\right] \tag{16} \end{equation*}(16)‖f‖L=1=∣[xk,xk+1;f]]
for k = 0 , 1 , … , n − 1 k = 0 , 1 , … , n − 1 k=0,1,dots,n-1k=0,1, \ldots, n-1k=0,1,…,n−1.
Proof. Suppose that relation (16) do not hold and let k ∈ { 0 , 1 , … , n − 1 } k ∈ { 0 , 1 , … , n − 1 } kin{0,1,dots,n-1}\mathbb{k} \in\{0,1, \ldots, n-1\}k∈{0,1,…,n−1} and ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 be such that
| [ x k , x k + 1 ; f ] | = 1 − ε . x k , x k + 1 ; f = 1 − ε . |[x_(k),x_(k+1);f]|=1-epsi.\left|\left[x_{k}, x_{k+1} ; f\right]\right|=1-\varepsilon .|[xk,xk+1;f]|=1−ε.
Let
f 1 ( x ) = { f ( x ) , x ∈ { x 0 , x 1 , … , x k } f ( x ) + δ , x ∈ { x k + 1 , … , x n } f 1 ( x ) = f ( x ) ,      x ∈ x 0 , x 1 , … , x k f ( x ) + δ ,      x ∈ x k + 1 , … , x n f_(1)(x)={[f(x)",",x in{x_(0),x_(1),dots,x_(k)}],[f(x)+delta",",x in{x_(k+1),dots,x_(n)}]:}f_{1}(x)= \begin{cases}f(x), & x \in\left\{x_{0}, x_{1}, \ldots, x_{k}\right\} \\ f(x)+\delta, & x \in\left\{x_{k+1}, \ldots, x_{n}\right\}\end{cases}f1(x)={f(x),x∈{x0,x1,…,xk}f(x)+δ,x∈{xk+1,…,xn}
and
f 2 ( x ) = { f ( x ) , x ∈ { x 0 , x 1 , … , x k } f ( x ) − δ , x ∈ { x k + 1 , … , x n } f 2 ( x ) = f ( x ) ,      x ∈ x 0 , x 1 , … , x k f ( x ) − δ ,      x ∈ x k + 1 , … , x n f_(2)(x)={[f(x)",",x in{x_(0),x_(1),dots,x_(k)}],[f(x)-delta",",x in{x_(k+1),dots,x_(n)}]:}f_{2}(x)= \begin{cases}f(x), & x \in\left\{x_{0}, x_{1}, \ldots, x_{k}\right\} \\ f(x)-\delta, & x \in\left\{x_{k+1}, \ldots, x_{n}\right\}\end{cases}f2(x)={f(x),x∈{x0,x1,…,xk}f(x)−δ,x∈{xk+1,…,xn}
where δ = ( ε / 2 ) ( x k + 1 − x k ) δ = ( ε / 2 ) x k + 1 − x k delta=(epsi//2)(x_(k+1)-x_(k))\delta=(\varepsilon / 2)\left(x_{k+1}-x_{k}\right)δ=(ε/2)(xk+1−xk).
Because
| f ( x k + 1 ) + δ − f ( x k ) | x k + 1 − x k ⩽ 1 − ε / 2 < 1 and | f ( x k + 1 ) − δ − f ( x k ) | x k + 1 − x k ⩽ 1 − ε / 2 < 1 , it follows that f x k + 1 + δ − f x k x k + 1 − x k ⩽ 1 − ε / 2 < 1  and  f x k + 1 − δ − f x k x k + 1 − x k ⩽ 1 − ε / 2 < 1 , it follows that  {:[(|f(x_(k+1))+delta-f(x_(k))|)/(x_(k+1)-x_(k)) <= 1-epsi//2 < 1quad" and "],[(|f(x_(k+1))-delta-f(x_(k))|)/(x_(k+1)-x_(k)) <= 1-epsi//2 < 1", it follows that "]:}\begin{aligned} & \frac{\left|f\left(x_{k+1}\right)+\delta-f\left(x_{k}\right)\right|}{x_{k+1}-x_{k}} \leqslant 1-\varepsilon / 2<1 \quad \text { and } \\ & \frac{\left|f\left(x_{k+1}\right)-\delta-f\left(x_{k}\right)\right|}{x_{k+1}-x_{k}} \leqslant 1-\varepsilon / 2<1 \text {, it follows that } \end{aligned}|f(xk+1)+δ−f(xk)|xk+1−xk⩽1−ε/2<1 and |f(xk+1)−δ−f(xk)|xk+1−xk⩽1−ε/2<1, it follows that 
‖ f 1 ‖ L = 1 = ‖ f 2 ‖ J f 1 L = 1 = f 2 J ||f_(1)||_(L)=1=||f_(2)||_(J)\left\|f_{1}\right\|_{L}=1=\left\|f_{2}\right\|_{J}‖f1‖L=1=‖f2‖J. But f = ( 1 / 2 ) ( f 1 + f 2 ) f = ( 1 / 2 ) f 1 + f 2 f=(1//2)(f_(1)+f_(2))f=(1 / 2)\left(f_{1}+f_{2}\right)f=(1/2)(f1+f2), so that f f fff is not an extreme point of B ( Lip 0 M ) B Lip 0 M B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right)B(Lip0M).
Suppose now that condition (16) is fulfilled and there exist two functions g 1 , g 2 ∈ B ( Lip 0 M ) g 1 , g 2 ∈ B Lip 0 M g_(1),g_(2)in B(Lip_(0)M)g_{1}, g_{2} \in B\left(\operatorname{Lip}_{0} M\right)g1,g2∈B(Lip0M) such that g 1 ≠ f ≠ g 2 g 1 ≠ f ≠ g 2 g_(1)!=f!=g_(2)g_{1} \neq f \neq g_{2}g1≠f≠g2 and f = ( 1 / 2 ) ( g 1 + g 2 ) f = ( 1 / 2 ) g 1 + g 2 f=(1//2)(g_(1)+g_(2))f=(1 / 2)\left(g_{1}+g_{2}\right)f=(1/2)(g1+g2). Let x i x i x_(i)x_{i}xi be the smallest element of M M MMM for which g 1 ( x i ) ≠ f ( x i ) g 1 x i ≠ f x i g_(1)(x_(i))!=f(x_(i))g_{1}\left(x_{i}\right) \neq f\left(x_{i}\right)g1(xi)≠f(xi). As g 1 ( x 0 ) = f ( x 0 ) = 0 g 1 x 0 = f x 0 = 0 g_(1)(x_(0))=f(x_(0))=0g_{1}\left(x_{0}\right)=f\left(x_{0}\right)=0g1(x0)=f(x0)=0 and g 2 ( x 0 ) = 0 == f ( x 0 ) g 2 x 0 = 0 == f x 0 g_(2)(x_(0))=0==f(x_(0))g_{2}\left(x_{0}\right)=0= =f\left(x_{0}\right)g2(x0)=0==f(x0), it follows i ⩾ 1 i ⩾ 1 i >= 1i \geqslant 1i⩾1.
Case I. f ( x i ) > f ( x i − 1 ) f x i > f x i − 1 f(x_(i)) > f(x_(i-1))f\left(x_{i}\right)>f\left(x_{i-1}\right)f(xi)>f(xi−1) and g 1 ( x i ) > f ( x i ) g 1 x i > f x i g_(1)(x_(i)) > f(x_(i))g_{1}\left(x_{i}\right)>f\left(x_{i}\right)g1(xi)>f(xi). In this case
g ( x i ) − g 1 ( x i − 1 ) x i − x i − 1 > f ( x i ) − f ( x i − 1 ) x i − x i − 1 = 1 g x i − g 1 x i − 1 x i − x i − 1 > f x i − f x i − 1 x i − x i − 1 = 1 (g(x_(i))-g_(1)(x_(i-1)))/(x_(i)-x_(i-1)) > (f(x_(i))-f(x_(i-1)))/(x_(i)-x_(i-1))=1\frac{g\left(x_{i}\right)-g_{1}\left(x_{i-1}\right)}{x_{i}-x_{i-1}}>\frac{f\left(x_{i}\right)-f\left(x_{i-1}\right)}{x_{i}-x_{i-1}}=1g(xi)−g1(xi−1)xi−xi−1>f(xi)−f(xi−1)xi−xi−1=1
which implies ‖ g 1 ‖ L > 1 g 1 L > 1 ||g_(1)||_(L) > 1\left\|g_{1}\right\|_{L}>1‖g1‖L>1, i.e. g 1 ∉ B ( Lip 0 M ) g 1 ∉ B Lip 0 M quadg_(1)!in B(Lip_(0)M)\quad g_{1} \notin B\left(\operatorname{Lip}_{0} M\right)g1∉B(Lip0M).
‖ E 1 ‖ L > g 1 ∉ B ( Lip M ) E 1 L > g 1 ∉ B ( Lip M ) ||E_(1)||_(L) > g_(1)!in B(Lip M)\left\|\mathrm{E}_{1}\right\|_{L}>\mathrm{g}_{1} \notin B(\operatorname{Lip} M)‖E1‖L>g1∉B(LipM). . . . . . .
Case II. f ( x i ) > f ( x i − 1 ) f x i > f x i − 1 f(x_(i)) > f(x_(i-1))f\left(x_{i}\right)>f\left(x_{i-1}\right)f(xi)>f(xi−1) and g 1 ( x i ) < f ( x i ) g 1 x i < f x i g_(1)(x_(i)) < f(x_(i))g_{1}\left(x_{i}\right)<f\left(x_{i}\right)g1(xi)<f(xi). In this case g 2 ( x 1 ) = 2 f ( x 1 ) − g 1 ( x i ) > f ( x i ) g 2 x 1 = 2 f x 1 − g 1 x i > f x i g_(2)(x_(1))=2f(x_(1))-g_(1)(x_(i)) > f(x_(i))g_{2}\left(x_{1}\right)=2 f\left(x_{1}\right)-g_{1}\left(x_{i}\right)>f\left(x_{i}\right)g2(x1)=2f(x1)−g1(xi)>f(xi) and
g 2 ( x i ) − g 2 ( x i − 1 ) x i − x i − 1 > f ( x i ) − [ 2 f ( x i − 1 ) − g 1 ( x i − 1 ) ] x i − x i − 1 = = f ( x i ) − f ( x i − 1 ) x i − x i − 1 = 1 g 2 x i − g 2 x i − 1 x i − x i − 1 > f x i − 2 f x i − 1 − g 1 x i − 1 x i − x i − 1 = = f x i − f x i − 1 x i − x i − 1 = 1 {:[(g_(2)(x_(i))-g_(2)(x_(i-1)))/(x_(i)-x_(i-1)) > (f(x_(i))-[2f(x_(i-1))-g_(1)(x_(i-1))])/(x_(i)-x_(i-1))=],[=(f(x_(i))-f(x_(i-1)))/(x_(i)-x_(i-1))=1]:}\begin{aligned} \frac{g_{2}\left(x_{i}\right)-g_{2}\left(x_{i-1}\right)}{x_{i}-x_{i-1}}> & \frac{f\left(x_{i}\right)-\left[2 f\left(x_{i-1}\right)-g_{1}\left(x_{i-1}\right)\right]}{x_{i}-x_{i-1}}= \\ & =\frac{f\left(x_{i}\right)-f\left(x_{i-1}\right)}{x_{i}-x_{i-1}}=1 \end{aligned}g2(xi)−g2(xi−1)xi−xi−1>f(xi)−[2f(xi−1)−g1(xi−1)]xi−xi−1==f(xi)−f(xi−1)xi−xi−1=1
so that ‖ g 2 ‖ L > 1 g 2 L > 1 ||g_(2)||_(L) > 1\left\|g_{2}\right\|_{L}>1‖g2‖L>1, i.e. g 2 ∉ B ( Iip p 0 M ) g 2 ∉ B Iip p 0 M g_(2)!in B(Iipp_(0)M)g_{2} \notin B\left(\operatorname{Iip} p_{0} M\right)g2∉B(Iipp0M).
In the remaining cases, i.e. f ( x i ) < f ( x i − 1 ) f x i < f x i − 1 f(x_(i)) < f(x_(i-1))f\left(x_{i}\right)<f\left(x_{i-1}\right)f(xi)<f(xi−1) and g 1 ( x i ) > f ( x i ) g 1 x i > f x i g_(1)(x_(i)) > f(x_(i))g_{1}\left(x_{i}\right)>f\left(x_{i}\right)g1(xi)>f(xi) or f ( x i ) < f ( x i − 1 ) f x i < f x i − 1 f(x_(i)) < f(x_(i-1))f\left(x_{i}\right)<f\left(x_{i-1}\right)f(xi)<f(xi−1) and g 1 ( x i ) < f ( x i ) g 1 x i < f x i g_(1)(x_(i)) < f(x_(i))g_{1}\left(x_{i}\right)<f\left(x_{i}\right)g1(xi)<f(xi), we have similarly ‖ g 2 ‖ L > 1 g 2 L > 1 ||g_(2)||_(L) > 1\left\|g_{2}\right\|_{L}>1‖g2‖L>1, respectively ‖ g 1 ‖ I > 1 g 1 I > 1 ||g_(1)||_(I) > 1\left\|g_{1}\right\|_{I}>1‖g1‖I>1. The obtained contradictions show that f f fff must be an extreme point of B ( Lip 0 M ) B Lip 0 M B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right)B(Lip0M).
3.2 Remark. Taking in (16) all possible signs, it follows that the unit ball of the space Lip 0 M 0 M _(0)M{ }_{0} M0M has exactly 2 n + 1 2 n + 1 2^(n+1)2^{n+1}2n+1 extreme points.
3.3 COROLLARY. Let f ∈ Lip [ a , b ] , M = { x 0 , x 1 , … , x n } f ∈ Lip [ a , b ] , M = x 0 , x 1 , … , x n fin Lip[a,b],M={x_(0),x_(1),dots,x_(n)}\mathrm{f} \in \operatorname{\operatorname {Lip}}[\mathrm{a}, \mathrm{b}], \mathrm{M}=\left\{\mathrm{x}_{0}, \mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right\}f∈Lip[a,b],M={x0,x1,…,xn}, a = x 0 < x 1 < … < x n = b a = x 0 < x 1 < … < x n = b a=x_(0) < x_(1) < dots < x_(n)=ba=x_{0}<x_{1}<\ldots<x_{n}=ba=x0<x1<…<xn=b, and f ( x 0 ) = 0 f x 0 = 0 f(x_(0))=0f\left(x_{0}\right)=0f(x0)=0. If
‖ f ‖ L = I = | [ x k , x k + 1 ; f ] | ‖ f ‖ L = I = x k , x k + 1 ; f ||f||_(L)=I=|[x_(k),x_(k+1);f]|\|f\|_{L}=I=\left|\left[x_{k}, x_{k+1} ; f\right]\right|‖f‖L=I=|[xk,xk+1;f]|
for all k = 0 , 1 , … , n − 1 k = 0 , 1 , … , n − 1 k=0,1,dots,n-1k=0,1, \ldots, n-1k=0,1,…,n−1, then f f fff is an extreme point of the unit ball of Lip o [ a , b ] Lip o [ a , b ] Lip_(o)[a,b]\operatorname{Lip}_{\mathrm{o}}[\mathrm{a}, \mathrm{b}]Lipo[a,b].
3.4 Remark. If f f fff is an extreme point of B ( Lip 0 M ) B Lip 0 M B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right)B(Lip0M) and the set M M MMM is as in Corollary 3.3 , then the unique Lipschitz extension F F FFF of f , F ( x ) = sup { f ( x i ) − | x − x i | : i = 0 , 1 , … , n } = inf { f ′ ( x i ) + | x − x i | : i = 0 , 1 , … , n } , x ∈ [ a , b ] f , F ( x ) = sup f x i − x − x i : i = 0 , 1 , … , n = inf f ′ x i + x − x i : i = 0 , 1 , … , n , x ∈ [ a , b ] f,F(x)=s u p{f(x_(i))-|x-x_(i)|:i=0,1,dots,n}=i n f{f^(')(x_(i))+|x-x_(i)|:i=0,1,dots,n},x in[a,b]f, F(x)=\sup \left\{f\left(x_{i}\right)-\left|x-x_{i}\right|: i=0,1, \ldots, n\right\} =\inf \left\{f^{\prime}\left(x_{i}\right)+\left|x-x_{i}\right|: i=0,1, \ldots, n\right\}, x \in[a, b]f,F(x)=sup{f(xi)−|x−xi|:i=0,1,…,n}=inf{f′(xi)+|x−xi|:i=0,1,…,n},x∈[a,b], is an extreme point of B ( Lip 0 [ a , b ] ) B Lip 0 [ a , b ] B(Lip_(0)[a,b])B\left(\operatorname{Lip}_{0}[a, b]\right)B(Lip0[a,b]).

4. Best approximation of Lipschitz functions.

Let
M = { x 0 , x 1 , … , x n } , a ≤ x 0 < x 1 < … < x n ≤ b M = x 0 , x 1 , … , x n , a ≤ x 0 < x 1 < … < x n ≤ b M={x_(0),x_(1),dots,x_(n)}quad,quad a <= x_(0) < x_(1) < dots < x_(n) <= bM=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\} \quad, \quad a \leq x_{0}<x_{1}<\ldots<x_{n} \leq bM={x0,x1,…,xn},a≤x0<x1<…<xn≤b
and let
(17) Lip 0 [ a , b ] = { f , f : [ a , b ] → R , f Lip 0 [ a , b ] = f , f : [ a , b ] → R , f quadLip_(0)[a,b]={f,f:[a,b]rarr R,f:}\quad \operatorname{Lip}_{0}[a, b]=\left\{f, f:[a, b] \rightarrow R, f\right.Lip0[a,b]={f,f:[a,b]→R,f is Lipschitz and f ( x 0 ) = 0 } f x 0 = 0 {:f(x_(0))=0}\left.f\left(x_{0}\right)=0\right\}f(x0)=0}. Let also
M ⊥ = { f ∈ Li p o [ a , b ] ; f | M = 0 } M ⊥ = f ∈ Li p o [ a , b ] ; f M = 0 M^(_|_)={f in Lip_(o)[a,b];f|_(M)=0}M^{\perp}=\left\{f \in \operatorname{Li} p_{o}[a, b] ;\left.f\right|_{M}=0\right\}M⊥={f∈Lipo[a,b];f|M=0}
Since every function f ∈ Lip 0 M f ∈ Lip 0 M f inLip_(0)Mf \in \operatorname{Lip}_{0} Mf∈Lip0M has at least one extension F ∈ Lip 0 [ a , b ] F ∈ Lip 0 [ a , b ] F inLip_(0)[a,b]F \in \operatorname{Lip}_{0}[a, b]F∈Lip0[a,b] it follows that every function f ∈ Lip 0 [ a , b ] f ∈ Lip 0 [ a , b ] f inLip_(0)[a,b]f \in \operatorname{Lip}_{0}[a, b]f∈Lip0[a,b] has a best approximation (nearest point) in M ⊥ M ⊥ M^(_|_)M^{\perp}M⊥, i.e. there exists E 0 ∈ M ⊥ E 0 ∈ M ⊥ E_(0)inM^(_|_)\mathrm{E}_{0} \in \mathbb{M}^{\perp}E0∈M⊥. such that ‖ f − g 0 ‖ L = inf { ‖ P − g ‖ L : g ∈ M ⊥ } f − g 0 L = inf ‖ P − g ‖ L : g ∈ M ⊥ ||f-g_(0)||_(L)=i n f{||P-g||_(L):g inM^(_|_)}\left\|\mathrm{f}-\mathrm{g}_{0}\right\|_{\mathrm{L}}=\inf \left\{\|\mathrm{P}-\mathrm{g}\|_{\mathrm{L}}: g \in \mathbb{M}^{\perp}\right\}‖f−g0‖L=inf{‖P−g‖L:g∈M⊥}. It Was shown ( see [5] ) that g 0 ∈ M ⊥ g 0 ∈ M ⊥ g_(0)inM^(_|_)g_{0} \in M^{\perp}g0∈M⊥ is an element of best approximation for f ∈ Lip 0 [ a , b ] f ∈ Lip 0 [ a , b ] f inLip_(0)[a,b]f \in \operatorname{Lip}_{0}[a, b]f∈Lip0[a,b] by elements from M ⊥ M ⊥ M^(_|_)M^{\perp}M⊥ if and only if C 0 = f − F C 0 = f − F C_(0)=f-FC_{0}=f-FC0=f−F, for F ∈ V ( f ) F ∈ V ( f ) F in V(f)F \in V(f)F∈V(f).
Taking into account the precedings results it follows :
4.1 THEOREIM. Let f ∈ Lip p 0 [ a , b ] f ∈ Lip p 0 [ a , b ] f in Lipp_(0)[a,b]f \in \operatorname{Lip} p_{0}[a, b]f∈Lipp0[a,b] and M = { x 0 , x 1 , … , x n } M = x 0 , x 1 , … , x n M={x_(0),x_(1),dots,x_(n)}M=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\}M={x0,x1,…,xn}, a ⩽ x 0 < x 1 < … < x n ⩽ b a ⩽ x 0 < x 1 < … < x n ⩽ b a <= x_(0) < x_(1) < dots < x_(n) <= ba \leqslant x_{0}<x_{1}<\ldots<x_{n} \leqslant ba⩽x0<x1<…<xn⩽b.
(a) If the relation (15) holds for an index k ∈ { 0 , 1 , … , n − 1 } k ∈ { 0 , 1 , … , n − 1 } k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\}k∈{0,1,…,n−1} then all the best approximation elements for f f fff in M ⊥ M ⊥ M^(_|_)M^{\perp}M⊥ vanish on the interval [ x k , x k + 1 ] x k , x k + 1 [x_(k),x_(k+1)]\left[\mathrm{x}_{\mathrm{k}}, \mathrm{x}_{\mathrm{k}+1}\right][xk,xk+1];
(b) If x 0 = a , x n = b x 0 = a , x n = b x_(0)=a,x_(n)=bx_{0}=a, x_{n}=bx0=a,xn=b and ‖ f ‖ L = | [ x 0 , x n ; f ] | ‖ f ‖ L = x 0 , x n ; f ||f||_(L)=|[x_(0),x_(n);f]|\|f\|_{L}=\left|\left[x_{0}, x_{n} ; f\right]\right|‖f‖L=|[x0,xn;f]|, then 0 is the only best approximation element for f f fff in M M MMM;
(c) If ‖ f ‖ L = | [ x k , x k + 1 ; f ] | ‖ f ‖ L = x k , x k + 1 ; f ||f||_(L)=|[x_(k),x_(k+1);f]|\|f\|_{L}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|‖f‖L=|[xk,xk+1;f]|, for all k = 0 , 1 , … , n − 1 k = 0 , 1 , … , n − 1 k=0,1,dots,n-1k=0,1, \ldots, n-1k=0,1,…,n−1, then 0 is the only best approximation olement for it in u ⊥ u ⊥ u^(_|_)u^{\perp}u⊥.
Proof. Assertion (a) follows from Corollary 2.3 , taking into account that the best approximation elements g 0 g 0 g_(0)g_{0}g0 for f f fff in ii ⊥ ⊥ ^(_|_){ }^{\perp}⊥ have the form
g 0 = I − E , F ∈ V ( P ) g 0 = I − E , F ∈ V ( P ) g_(0)=I-E quad,quad F inV(P)g_{0}=I-E \quad, \quad F \in \mathbb{V}(P)g0=I−E,F∈V(P)
Assertions (b) and (c) follow from Corollary 2.4 .
4.2 Remark. Let M M MMM be the set (8) and let
Lip 0 [ a , b ] = { f : [ a , b ] → R , f ( x 0 ) = 0 , f is Lipschitz } Lip 0 [ a , b ] = f : [ a , b ] → R , f x 0 = 0 , f  is Lipschitz  Lip_(0)[a,b]={f:[a,b]rarr R,f(x_(0))=0,f" is Lipschitz "}\operatorname{Lip}_{0}[a, b]=\left\{f:[a, b] \rightarrow R, f\left(x_{0}\right)=0, f \text { is Lipschitz }\right\}Lip0[a,b]={f:[a,b]→R,f(x0)=0,f is Lipschitz }
If f ∉ W ( f ) f ∉ W ( f ) f!in W(f)f \notin W(f)f∉W(f), then, by Theorem 2.1, it follws
‖ f ‖ L > ‖ f | M ‖ L ‖ f ‖ L > f M L ||f||_(L) > ||f|_(M)||_(L)\|f\|_{L}>\left\|\left.f\right|_{M}\right\|_{L}‖f‖L>‖f|M‖L
If, further
(i)
f ( x ) ⩽ F 1 ( x ) , x ∈ [ a , b ] f ( x ) ⩽ F 1 ( x ) , x ∈ [ a , b ] f(x) <= F_(1)(x)quad,quad x in[a,b]f(x) \leqslant F_{1}(x) \quad, \quad x \in[a, b]f(x)⩽F1(x),x∈[a,b]
where F 1 F 1 F_(1)F_{1}F1 is given by (5) (with A = M A = M A=MA=MA=M ), then all bect approximation elements of f f fff in M ⊥ M ⊥ M^(_|_)M^{\perp}M⊥ are non-positive, and if
(ii) f ( x ) ⩾ F 2 ( x ) , x ∈ [ a , b ] (ii) f ( x ) ⩾ F 2 ( x ) , x ∈ [ a , b ] {:(ii)f(x) >= F_(2)(x)quad","quad x in[a","b]:}\begin{equation*} f(x) \geqslant F_{2}(x) \quad, \quad x \in[a, b] \tag{ii} \end{equation*}(ii)f(x)⩾F2(x),x∈[a,b]
with F 2 F 2 F_(2)F_{2}F2 given by (6) (with A = M A = M A=MA=MA=M ), then all the elements of best approximation for f f fff in M ⊥ M ⊥ M^(_|_)M^{\perp}M⊥ are non-negative.
Obviously, there exist functions f ∈ Lip o [ a , b ] f ∈ Lip o [ a , b ] f inLip_(o)[a,b]f \in \operatorname{Lip}_{o}[a, b]f∈Lipo[a,b] verifying the conditions (i) and (ii) and which are not fixed points of W W WWW.
For example, takin G g ∈ Lip o [ a , b ] G g ∈ Lip o [ a , b ] _(G)g inLip_(o)[a,b]{ }_{G} g \in \operatorname{Lip}_{o}[a, b]Gg∈Lipo[a,b], the function
f ( x ) = sup { g ( x k ) − L i | x − x k | : x k ∈ M , k = 0 , 1 , … , n } f ( x ) = sup g x k − L i x − x k : x k ∈ M , k = 0 , 1 , … , n f(x)=s u p{g(x_(k))-L_(i)|x-x_(k)|:x_(k)in M,k=0,1,dots,n}f(x)=\sup \left\{g\left(x_{k}\right)-L_{i}\left|x-x_{k}\right|: x_{k} \in M, k=0,1, \ldots, n\right\}f(x)=sup{g(xk)−Li|x−xk|:xk∈M,k=0,1,…,n}
with L > ‖ g | M ‖ L L > g M L L > ||g|_(M)||_(L)L>\left\|\left.g\right|_{\mathbb{M}}\right\|_{L}L>‖g|M‖L, verifies the condition (i) and the function
h ( x ) = inf { g ( x k ) + I . | x − x k | : x k ∈ h , k = 0 , 1 , … , n } h ( x ) = inf g x k + I . x − x k : x k ∈ h , k = 0 , 1 , … , n h(x)=i n f{g(x_(k))+I.|x-x_(k)|:x_(k)in h,k=0,1,dots,n}h(x)=\inf \left\{g\left(x_{k}\right)+I .\left|x-x_{k}\right|: x_{k} \in h, k=0,1, \ldots, n\right\}h(x)=inf{g(xk)+I.|x−xk|:xk∈h,k=0,1,…,n}
with L > ‖ B | ‖ ‖ I L > B ‖ I L > ||B|_(||)||_(I)L>\left\|\left.B\right|_{\|}\right\|_{I}L>‖B|‖‖I verifies the condition h ( x ) ⩽ I 2 ( x ) h ( x ) ⩽ I 2 ( x ) h(x) <= I_(2)(x)h(x) \leqslant I_{2}(x)h(x)⩽I2(x), x ∈ [ a , b ] x ∈ [ a , b ] x in[a,b]x \in[a, b]x∈[a,b] (i.e. condition (ii) ) .
1 COBZAS, S. , MUSTATA, C. , Norm Preserving Extension of Convex Iipschitz Functions, J.A.T. 24 (1978), 236 - 244.
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4 Wic SHATL, E.J., Extension of range of functions, Bull. Amer. Jath. Soc. 40 (1934), 837-842.
5 USTATA, C., Best Approximation and Unique Intiension of Lipsch itz Functions , J.A.T. 19 (1977), 222-230.
6 R.OI, A.I., Extreme Points and Iinear Isometries of Banach Space of Lijschitz Functions, Canad. J. of Math. 20 (1968), 1150-1164.
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Acad. R.S. România, Bucureşti, 1967 .
1986

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