On the selections associated to the metric projection

Costica Mustata
(original), Approximation Theory, best approximation, paper

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

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C. Mustăţa, On the selections associated to the metric projection, Rev. Anal. Numér. Théor. Approx. 23 (1994) 1, 89-93 (MR # 96b: 46019)

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https://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art9/Mustata

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Revue d’Analyse Numer. Theor. Approx.

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Publishing Romanian Academy

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https://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art9/

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2501-059X

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2457-6794

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[1] Deutsch, F., Linear Selections for the Metric Projection, J. Funct. Analysis 49(1982) 269-292.
[2] Deutsch, F., Wu Li, Sung-Ho Park, Tietze Extensions and Continuous Selections for Metric  Projections, J.A.T., 64 (1991), 55-68.
[3] Mustata, C., Selections Associated to the Mc Shane’s Extension Theorem for Lipschitz Functions, Revue d’Analyse Numer et de Theorie de l’Approximtion 21 (2) (1992), 135-145.
[4] Mustata, C., Best Approximation and Unique Extension of Lipschitz Functions, J.A.T. 19 (1977), 222-230.
[5] Mustata, C., A Characterization of Chebyshevian Subspace of  Y^{⊥}- type Mathematica – Revue d’Analyse Numer. et de Theorie de l’Approximation, 6 (1) (1977), 51-56.
[6] Singer, I., Best Approximation in Vector Normed Spaces by Elements of Vector Subspaces (in Romanian) Ed. Acad., Bucuresti, 1967.

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1994-Mustata-On the selections associated to the metric projection-Jnaat

ON THE SELECTIONS ASSOCIATED TO THE METRIC PROJECTION

COSTICA MUSTATA(Chuj-Napoca)

1. INTRODUCTION

Let X X XXX be a real normed space and M M MMM a closed subspace of X X XXX. The distance from a point x ∈ X x ∈ X x in Xx \in Xx∈X to M M MMM is defined by
(1) d ( x , M ) := inf { ‖ x − y ‖ : y ∈ M } . (1) d ( x , M ) := inf { ‖ x − y ‖ : y ∈ M } . {:(1)d(x","M):=i n f{||x-y||:y in M}.:}\begin{equation*} d(x, M):=\inf \{\|x-y\|: y \in M\} . \tag{1} \end{equation*}(1)d(x,M):=inf{‖x−y‖:y∈M}.
Let
(2) P M ( x ) := { y ∈ M : ‖ x − y ‖ = d ( x , M ) } (2) P M ( x ) := { y ∈ M : ‖ x − y ‖ = d ( x , M ) } {:(2)P_(M)(x):={y in M:||x-y||=d(x","M)}:}\begin{equation*} P_{M}(x):=\{y \in M:\|x-y\|=d(x, M)\} \tag{2} \end{equation*}(2)PM(x):={y∈M:‖x−y‖=d(x,M)}
be the set of elements of best approximation for x x xxx in M M MMM. The subspace M M MMM is called proximinal (Chebyshevian) if P M ( x ) ≠ ∅ P M ( x ) ≠ ∅ P_(M)(x)!=O/P_{M}(x) \neq \varnothingPM(x)≠∅ (respectively P M ( x ) P M ( x ) P_(M)(x)P_{M}(x)PM(x) is a singleton), for all x ∈ X x ∈ X x in Xx \in Xx∈X. If M M MMM is a proximinal subspace of X X XXX, the multivalued operator P M : X → 2 M P M : X → 2 M P_(M):X rarr2^(M)P_{M}: X \rightarrow 2^{M}PM:X→2M is called the metric projection of X X XXX onto M M MMM.
An application p : X − M p : X − M p:X-Mp: X-Mp:X−M such that p ( x ) ∈ P M ( x ) p ( x ) ∈ P M ( x ) p(x)inP_(M)(x)p(x) \in P_{M}(x)p(x)∈PM(x), for all x ∈ X x ∈ X x in Xx \in Xx∈X, is called a selection for P M P M P_(M)\boldsymbol{P}_{M}PM.
The set
(3) Ker P M := { x ∈ X : θ ∈ P M ( x ) } (3) Ker P M := x ∈ X : θ ∈ P M ( x ) {:(3)KerP_(M):={x in X:theta inP_(M)(x)}:}\begin{equation*} \operatorname{Ker} P_{M}:=\left\{x \in X: \theta \in P_{M}(x)\right\} \tag{3} \end{equation*}(3)KerPM:={x∈X:θ∈PM(x)}
is called the kernel of the metric projection.
If K K KKK is a subset; of X X XXX and M M MMM is proximinal (Chebyshevian) only for the elements of K K KKK, then the subspace M M MMM is called K K KKK-proximinal (respectively K K KKK-Chebyshevian).
The restriction of the metric projection P M P M P_(M)P_{M}PM to K K KKK is denoted by P M / M P M / M P_(M//M)P_{M / M}PM/M
(4) K e x P M / M := { x ∈ K : θ ∈ P M ( x ) } (4) K e x P M / M := x ∈ K : θ ∈ P M ( x ) {:(4)KexP_(M//M):={x in K:theta inP_(M)(x)}:}\begin{equation*} K e x \mathbb{P}_{M / M}:=\left\{x \in K: \theta \in P_{M}(x)\right\} \tag{4} \end{equation*}(4)KexPM/M:={x∈K:θ∈PM(x)}
is called the kernel of the metric projection relative to K K KKK.
For two subsets U , V U , V U,VU, VU,V of X X XXX, their sum is defined by U + V :== { u + v : u ∈ U , v ∈ V } U + V :== { u + v : u ∈ U , v ∈ V } U+V:=={u+v:u in U,v in V}U+V:= =\{u+v: u \in U, v \in V\}U+V:=={u+v:u∈U,v∈V}. If eyery x ∈ U + V x ∈ U + V x in U+Vx \in U+Vx∈U+V can be uniquely written in the form x = u + v x = u + v x=u+vx=u+vx=u+v, for u ∈ U u ∈ U u in Uu \in Uu∈U and v ∈ V v ∈ V v in Vv \in Vv∈V, then this sum is called the algebraic direct sum of the sets U U UUU and V V VVV and is denoted by U ∗ + V U ∗ + V U^(**)+VU^{*}+VU∗+V. If K == U + V K == U + V K==U+VK= =U+VK==U+V and the application ( u , v ) → u + v ( u , v ) → u + v (u,v)rarr u+v(u, v) \rightarrow u+v(u,v)→u+v is a topological homeonorphism between U × V U × V U xx VU \times VU×V and K K KKK, then K K KKK is called the topological direct sum of the set U U UUU and V V VVV and is denoted by K = U ⊕ V K = U ⊕ V K=U o+VK=U \oplus VK=U⊕V.

MAIN THEOREM

Obviously that for K ⊂ X K ⊂ X K sub XK \subset XK⊂X and a K K KKK-proximinal subspace M M MMM of X X XXX, the properties of the selection associated to the metric projection P M / K P M / K P_(M//K)P_{M / K}PM/K depend, on one side, on the properties of the subspace M M MMM and on the other side, on the properties of the set K K KKK.
F. Deutsch [1] characterized the proximinal subspace M M MMM of X X XXX for which the metric projection P M P M P_(M)P_{M}PM admits a linear and continuous selection : namely, P M P M P_(M)P_{M}PM admits a continuous linear selection if and only if Ker P M P M P_(M)P_{M}PM contains a closed subspace complementary to M M MMM (see [1], Theorem 2.2).
The aim of this paper is to answer the following question : For a closed cone K K KKK in X X XXX and a K K KKK-proximinal subspace M M MMM of X X XXX when does the metric projection P M / K P M / K P_(M//K)P_{M / K}PM/K admit a continuous, additive and positively homogeneous selection?
A cone is a nonvoid subset K K KKK of X X XXX such that : a a aaa ) x + y ∈ K x + y ∈ K x+y in Kx+y \in Kx+y∈K, for all x , y ∈ K i ; b ) λ x ∈ K x , y ∈ K i ; b λ x ∈ K {:x,y inK_(i);b)lambda x in K\left.x, y \in K_{i} ; b\right) \lambda x \in Kx,y∈Ki;b)λx∈K, for all λ ⩾ 0 λ ⩾ 0 lambda >= 0\lambda \geqslant 0λ⩾0 and x ∈ K x ∈ K x inKx \in \mathbb{K}x∈K.
A partial answer to this question is suggestod by the above quoted result of F. Deutsch and is given in the following:
Theorem A. Let K K KKK be a closed cone in X X XXX and M M MMM a K K KKK-proximinal subspace of X X XXX. If the subspace M M MMM contains a dosed cone U U UUU and Ker P a / a P a / a P_(a//a)P_{a / a}Pa/a contains a closed cone C C CCC such that
(5) K ˙ = U ⊕ C , (5) K ˙ = U ⊕ C , {:(5)K^(˙)=U o+C",":}\begin{equation*} \dot{K}=U \oplus C, \tag{5} \end{equation*}(5)K˙=U⊕C,
then P M / K P M / K P_(M//K)P_{M / K}PM/K admits a positively homogeneous, additive and continuous selection.
Proof. Let U U UUU be a closed cone in M M MMM and C C CCC a closed cone in K o r P 1 / π K o r P 1 / π KorP_(1//pi)K o r P_{1 / \pi}KorP1/π such that K = U ⊕ C K = U ⊕ C K=U o+CK=U \oplus CK=U⊕C.
Then, for h ∈ K h ∈ K h in Kh \in Kh∈K, there exist uniquely devermined elements u h ∈ U u h ∈ U u_(h)in Uu_{h} \in Uuh∈U and c h ∈ C c h ∈ C c_(h)in Cc_{h} \in Cch∈C such that h = u h + c h h = u h + c h h=u_(h)+c_(h)h=u_{h}+c_{h}h=uh+ch.
Define the application q : K → M I q : K → M I q:K rarr MIq: K \rightarrow M Iq:K→MI by
q ( h ) = u h , h ∈ K . q ( h ) = u h , h ∈ K . q(h)=u_(h),quad h inK.q(h)=u_{h}, \quad h \in \mathbb{K} .q(h)=uh,h∈K.
Since K K KKK is homeomorph to U × O U × O U xx OU \times OU×O it follows that the application q q qqq is continuous.
Let h 1 , h 2 ∈ K h 1 , h 2 ∈ K h_(1),h_(2)in Kh_{1}, h_{2} \in Kh1,h2∈K and u h 1 , u h 2 ∈ U u h 1 , u h 2 ∈ U u_(h_(1)),u_(h_(2))in Uu_{h_{1}}, u_{h_{2}} \in Uuh1,uh2∈U and c h 1 , e h 2 ∈ C c h 1 , e h 2 ∈ C c_(h_(1)),e_(h_(2))in Cc_{h_{1}}, e_{h_{2}} \in Cch1,eh2∈C be such that h 1 == u h 1 + e h 1 , h 2 = u h 1 + e h 2 h 1 == u h 1 + e h 1 , h 2 = u h 1 + e h 2 h_(1)==u_(h_(1))+e_(h_(1)),h_(2)=u_(h_(1))+e_(h_(2))h_{1}= =u_{h_{1}}+e_{h_{1}}, h_{2}=u_{h_{1}}+e_{h_{2}}h1==uh1+eh1,h2=uh1+eh2. Then
q ( h 1 + h 2 ) = u h 1 + u h 2 = q ( h 1 ) + q ( h 2 ) , q h 1 + h 2 = u h 1 + u h 2 = q h 1 + q h 2 , q(h_(1)+h_(2))=u_(h_(1))+u_(h_(2))=q(h_(1))+q(h_(2)),q\left(h_{1}+h_{2}\right)=u_{h_{1}}+u_{h_{2}}=q\left(h_{1}\right)+q\left(h_{2}\right),q(h1+h2)=uh1+uh2=q(h1)+q(h2),
showing that the application q q qqq is additive.
Also, for λ ⩾ 0 λ ⩾ 0 lambda >= 0\lambda \geqslant 0λ⩾0 and h ∈ K h ∈ K h in Kh \in Kh∈K it follows λ h ∈ K λ h ∈ K lambda h in K\lambda h \in Kλh∈K and
q ( λ h ) = λ u t h = λ q ( h ) , q ( λ h ) = λ u t h = λ q ( h ) , q(lambda h)=lambdau_(th)=lambda q(h),q(\lambda h)=\lambda u_{t h}=\lambda q(h),q(λh)=λuth=λq(h),
showing that q q qqq is positively homogeneous.
The inclusion C ⊆ C ⊆ C subeC \subseteqC⊆ Ker P a / K P a / K P_(a//K)P_{a / K}Pa/K implies that for h ∈ K , h = u h + c h ′ h ∈ K , h = u h + c h ′ h in K,h=u_(h)+c_(h^('))h \in K, h=u_{h}+c_{h^{\prime}}h∈K,h=uh+ch′, u h ∈ U , c h ∈ C u h ∈ U , c h ∈ C u_(h)in U,c_(h)in Cu_{h} \in U, c_{h} \in Cuh∈U,ch∈C, we have
u h = q ( h ) = q ( u h + c h ) = q ( u h ) + q ( c h ) = q ( u h ) u h = q ( h ) = q u h + c h = q u h + q c h = q u h u_(h)=q(h)=q(u_(h)+c_(h))=q(u_(h))+q(c_(h))=q(u_(h))u_{h}=q(h)=q\left(u_{h}+c_{h}\right)=q\left(u_{h}\right)+q\left(c_{h}\right)=q\left(u_{h}\right)uh=q(h)=q(uh+ch)=q(uh)+q(ch)=q(uh)
and therefore
‖ h − q ( h ) ‖ = ‖ u h + c h − u h ‖ = ‖ c h ‖ = d ( c h , M ) . ‖ h − q ( h ) ‖ = u h + c h − u h = c h = d c h , M . ||h-q(h)||=||u_(h)+c_(h)-u_(h)||=||c_(h)||=d(c_(h),M).\|h-q(h)\|=\left\|u_{h}+c_{h}-u_{h}\right\|=\left\|c_{h}\right\|=d\left(c_{h}, M\right) .‖h−q(h)‖=‖uh+ch−uh‖=‖ch‖=d(ch,M).
But
d ( u ′ + c h , M ) = d ( c h , M ) d u ′ + c h , M = d c h , M d(u^(')+c_(h),M)=d(c_(h),M)d\left(u^{\prime}+c_{h}, M\right)=d\left(c_{h}, M\right)d(u′+ch,M)=d(ch,M)
for every u ′ ∈ M u ′ ∈ M u^(')in Mu^{\prime} \in Mu′∈M, so that
‖ h − q ( h ) ‖ = d ( c k , M ) = d ( u k + c k , M ) = d ( h , M ) , ‖ h − q ( h ) ‖ = d c k , M = d u k + c k , M = d ( h , M ) , ||h-q(h)||=d(c_(k),M)=d(u_(k)+c_(k),M)=d(h,M),\|h-q(h)\|=d\left(c_{k}, M\right)=d\left(u_{k}+c_{k}, M\right)=d(h, M),‖h−q(h)‖=d(ck,M)=d(uk+ck,M)=d(h,M),
which shows that q ( h ) q ( h ) q(h)q(h)q(h) is a best approximation element for h h hhh in M M MMM.
In conclusion, the application q : K → M q : K → M q:K rarr Mq: K \rightarrow Mq:K→M is an additive, positively homogeneous and continuous selection of P M/K P M/K  P_("M/K ")P_{\text {M/K }}PM/K 

APPLICATIONS

1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ Let X := Lip 0 [ 0 , 1 ] X := Lip 0 [ 0 , 1 ] X:=Lip_(0)[0,1]X:=\operatorname{Lip}_{0}[0,1]X:=Lip0[0,1] be the linear space
(6) Lip [ 0 , 1 ] := { f ∣ f : [ 0 , 1 ] → R , f Lip [ 0 , 1 ] := { f ∣ f : [ 0 , 1 ] → R , f Lip[0,1]:={f∣f:[0,1]rarrR,f\operatorname{Lip}[0,1]:=\{f \mid f:[0,1] \rightarrow \mathbb{R}, fLip[0,1]:={f∣f:[0,1]→R,f is Lipschitz on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] and f ( 0 ) == 0 } f ( 0 ) == 0 } f(0)==0}f(0)= =0\}f(0)==0}, with the Lipschitz norm
(7) ‖ f ‖ L := sup { | f ( x ) − f ( y ) | | x − y | : x , y ∈ [ 0 , 1 ] , x ≠ y } (7) ‖ f ‖ L := sup | f ( x ) − f ( y ) | | x − y | : x , y ∈ [ 0 , 1 ] , x ≠ y {:(7)||f||_(L):=s u p{(|f(x)-f(y)|)/(|x-y|):x,y in[0,1],x!=y}:}\begin{equation*} \|f\|_{L}:=\sup \left\{\frac{|f(x)-f(y)|}{|x-y|}: x, y \in[0,1], x \neq y\right\} \tag{7} \end{equation*}(7)‖f‖L:=sup{|f(x)−f(y)||x−y|:x,y∈[0,1],x≠y}
Let
M := { g ∈ Lip 0 [ 0 , 1 ] : g ( 1 ) = g ( 0 ) = 0 } M := g ∈ Lip 0 [ 0 , 1 ] : g ( 1 ) = g ( 0 ) = 0 M:={g inLip_(0)[0,1]:g(1)=g(0)=0}M:=\left\{g \in \operatorname{Lip}_{0}[0,1]: g(1)=g(0)=0\right\}M:={g∈Lip0[0,1]:g(1)=g(0)=0}
K := { f ∈ Lip 0 [ 0 , 1 ] : f ( x ) ⩾ 0 , for all x ∈ [ 0 , 1 ] } . K := f ∈ Lip 0 [ 0 , 1 ] : f ( x ) ⩾ 0 ,  for all  x ∈ [ 0 , 1 ] . K:={f inLip_(0)[0,1]:f(x) >= 0," for all "x in[0,1]}.K:=\left\{f \in \operatorname{Lip}_{0}[0,1]: f(x) \geqslant 0, \text { for all } x \in[0,1]\right\} .K:={f∈Lip0[0,1]:f(x)⩾0, for all x∈[0,1]}.
Obviously that M M MMM is a K K KKK-proximinal closed subspace of Lip 0 [ 0 , 1 ] ∼ Lip 0 [ 0 , 1 ] ∼ Lip_(0)[0,1]∼\operatorname{Lip}_{0}[0,1] \simLip0[0,1]∼ if f ∈ K f ∈ K f in Kf \in Kf∈K then g 0 ∈ M g 0 ∈ M g_(0)in Mg_{0} \in Mg0∈M is an element of best approximation for f f fff if and only if g 0 = f − F g 0 = f − F g_(0)=f-Fg_{0}=f-Fg0=f−F, where F ( x ) = f ( 1 ) x , x ∈ [ 0 , 1 ] F ( x ) = f ( 1 ) x , x ∈ [ 0 , 1 ] F(x)=f(1)x,x in[0,1]F(x)=f(1) x, x \in[0,1]F(x)=f(1)x,x∈[0,1] ([4], Lemma 1).
In fact g 0 = f − F g 0 = f − F g_(0)=f-Fg_{0}=f-Fg0=f−F, with F F FFF given above, is a unique element of best approximation for f f fff in M M MMM, which means that M M MMM is a K K KKK-Chebyshevian subspace of Lip 0 [ 0 , 1 ] Lip 0 [ 0 , 1 ] Lip_(0)[0,1]\operatorname{Lip}_{0}[0,1]Lip0[0,1].
We have
‖ f − g 0 ‖ L = d ( f , M ) = inf { ‖ f − g ‖ L : g ∈ M } = f ( 1 ) f − g 0 L = d ( f , M ) = inf ‖ f − g ‖ L : g ∈ M = f ( 1 ) ||f-g_(0)||_(L)=d(f,M)=i n f{||f-g||_(L):g in M}=f(1)\left\|f-g_{0}\right\|_{L}=d(f, M)=\inf \left\{\|f-g\|_{L}: g \in M\right\}=f(1)‖f−g0‖L=d(f,M)=inf{‖f−g‖L:g∈M}=f(1)
and
Ker P u / K = { f ∈ K : ‖ f ‖ L = f ( 1 ) } = Ker P u / K = f ∈ K : ‖ f ‖ L = f ( 1 ) = KerP_(u//K)={f in K:||f||_(L)=f(1)}=\operatorname{Ker} P_{u / K}=\left\{f \in K:\|f\|_{L}=f(1)\right\}=KerPu/K={f∈K:‖f‖L=f(1)}=
= { h ∈ K : h ( x ) = α x , x ∈ [ 0 , 1 ] , α ⩾ 0 } = { h ∈ K : h ( x ) = α x , x ∈ [ 0 , 1 ] , α ⩾ 0 } ={h in K:h(x)=alpha x,x in[0,1],alpha >= 0}=\{h \in K: h(x)=\alpha x, x \in[0,1], \alpha \geqslant 0\}={h∈K:h(x)=αx,x∈[0,1],α⩾0}
In this case
C = Ker P u / κ C = Ker P u / κ C=Ker^(P_(u//kappa))C=\operatorname{Ker}^{P_{u / \kappa}}C=KerPu/κ
and every function f ∈ K f ∈ K f in Kf \in Kf∈K can be uniquely written in the form f = g + h f = g + h f=g+hf=g+hf=g+h with g ∈ U g ∈ U g in Ug \in Ug∈U and h ∈ C h ∈ C h in Ch \in Ch∈C, where h ( x ) = f ( 1 ) x , x ∈ [ 0 , 1 ] h ( x ) = f ( 1 ) x , x ∈ [ 0 , 1 ] h(x)=f(1)x,x in[0,1]h(x)=f(1) x, x \in[0,1]h(x)=f(1)x,x∈[0,1] and
U = { g ∣ g ( x ) = f ( x ) − f ( 1 ) x , x ∈ [ 0 , 1 ] , f ∈ K } C = { h ∣ h ( x ) = f ( 1 ) x , x ∈ [ 0 , 1 ] , f ∈ K } U = { g ∣ g ( x ) = f ( x ) − f ( 1 ) x , x ∈ [ 0 , 1 ] , f ∈ K } C = { h ∣ h ( x ) = f ( 1 ) x , x ∈ [ 0 , 1 ] , f ∈ K } {:[U={g∣g(x)=f(x)-f(1)x","x in[0","1]","f in K}],[C={h∣h(x)=f(1)x","x in[0","1]","f in K}]:}\begin{gathered} U=\{g \mid g(x)=f(x)-f(1) x, x \in[0,1], f \in K\} \\ C=\{h \mid h(x)=f(1) x, x \in[0,1], f \in K\} \end{gathered}U={g∣g(x)=f(x)−f(1)x,x∈[0,1],f∈K}C={h∣h(x)=f(1)x,x∈[0,1],f∈K}
Since M M MMM is K K KKK-Chebyshevian it follows that the metric projection operator P M / K P M / K P_(M//K)P_{M / K}PM/K is one-valued, continuous, additive and positively homogeneous.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ For x 0 ∈ ( 0 , 1 ) x 0 ∈ ( 0 , 1 ) x_(0)in(0,1)x_{0} \in(0,1)x0∈(0,1) fixed, consider the linear space
(10) X := Lip x 0 [ 0 , 1 ] = { f / f : [ 0 , 1 ] → R X := Lip x 0 [ 0 , 1 ] = { f / f : [ 0 , 1 ] → R X:=Lip_(x_(0))[0,1]={f//f:[0,1]rarrRX:=\operatorname{Lip}_{x_{0}}[0,1]=\{f / f:[0,1] \rightarrow \mathbb{R}X:=Lipx0[0,1]={f/f:[0,1]→R, f is Lipschitz on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] and f ( x 0 ) = 0 } f x 0 = 0 {:f(x_(0))=0}\left.f\left(x_{0}\right)=0\right\}f(x0)=0}, with the Lipschits norm (7).
Let
(11)
M := { g ∈ Lip x 0 [ 0 , 1 ] : g ( 0 ) = g ( x 0 ) = g ( 1 ) = 0 } , M := g ∈ Lip x 0 [ 0 , 1 ] : g ( 0 ) = g x 0 = g ( 1 ) = 0 , M:={g inLip_(x_(0))[0,1]:g(0)=g(x_(0))=g(1)=0},M:=\left\{g \in \operatorname{Lip}_{x_{0}}[0,1]: g(0)=g\left(x_{0}\right)=g(1)=0\right\},M:={g∈Lipx0[0,1]:g(0)=g(x0)=g(1)=0},
the annihilator in Lip x 0 [ 0 , 1 ] Lip x 0 [ 0 , 1 ] Lip_(x_(0))[0,1]\operatorname{Lip}_{x_{0}}[0,1]Lipx0[0,1] of the set { 0 , x 0 , 1 } 0 , x 0 , 1 {0,x_(0),1}\left\{0, x_{0}, 1\right\}{0,x0,1}, and
(12) K := { f ∈ Lip x x 0 [ 0 , 1 ] : f ( x ) ⩾ 0 K := f ∈ Lip x x 0 [ 0 , 1 ] : f ( x ) ⩾ 0 K:={f in Lipx_(x_(0))[0,1]:f(x) >= 0:}K:=\left\{f \in \operatorname{Lip} x_{x_{0}}[0,1]: f(x) \geqslant 0\right.K:={f∈Lipxx0[0,1]:f(x)⩾0, for all x ∈ [ 0 , 1 ] x ∈ [ 0 , 1 ] x in[0,1]x \in[0,1]x∈[0,1] ).
Again, M M MMM is a K K KKK-proximinal subspace of Lip x 0 [ 0 , 1 ] Lip x 0 [ 0 , 1 ] Lip_(x_(0))[0,1]\mathrm{Lip}_{x_{0}}[0,1]Lipx0[0,1].
Indeed, for f ∈ K f ∈ K f in Kf \in Kf∈K let
(13) E ( f ) = { F ∈ Iip x 0 [ 0 , 1 ] : F | { 0 , x 0 , 1 } = f | { 0 , x 0 , 1 } and ‖ F ′ ‖ L = max { f ( 0 ) x 0 , f ( 1 ) 1 − x 0 } (13) E ( f ) = F ∈ Iip x 0 [ 0 , 1 ] : F 0 , x 0 , 1 = f 0 , x 0 , 1  and  F ′ L = max f ( 0 ) x 0 , f ( 1 ) 1 − x 0 {:[(13)E(f)={F inIip_(x_(0))[0,1]:F|_({0,x_(0),1})=f|_({0,x_(0),1}):}" and "],[||F^(')||_(L)=max{(f(0))/(x_(0)),(f(1))/(1-x_(0))}]:}\begin{gather*} E(f)=\left\{F \in \operatorname{Iip}_{x_{0}}[0,1]:\left.F\right|_{\left\{0, x_{0}, 1\right\}}=\left.f\right|_{\left\{0, x_{0}, 1\right\}}\right. \text { and } \tag{13}\\ \left\|F^{\prime}\right\|_{L}=\max \left\{\frac{f(0)}{x_{0}}, \frac{f(1)}{1-x_{0}}\right\} \end{gather*}(13)E(f)={F∈Iipx0[0,1]:F|{0,x0,1}=f|{0,x0,1} and ‖F′‖L=max{f(0)x0,f(1)1−x0}
be the set of the extensions to [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] of the function f | { 0 , x 0 , 1 } f 0 , x 0 , 1 f|_({0,x_(0),1})\left.f\right|_{\left\{0, x_{0}, 1\right\}}f|{0,x0,1} which preserver the Lipschitz norm of f f fff on { 0 , x 0 , 1 } 0 , x 0 , 1 {0,x_(0),1}\left\{0, x_{0}, 1\right\}{0,x0,1}.
Then
(14)
and
P M / K ( f ) = f − E ( f ) P M / K ( f ) = f − E ( f ) P_(M//K)(f)=f-E(f)P_{M / K}(f)=f-E(f)PM/K(f)=f−E(f)
Ker P M / K = { f ∈ K : ‖ f ‖ L = max { f ( 0 ) x 0 , f ( 1 ) 1 − a ( j ) } } P M / K = f ∈ K : ‖ f ‖ L = max f ( 0 ) x 0 , f ( 1 ) 1 − a ( j ) P_(M//K)={f inK:||f||_(L)=max{(f(0))/(x_(0)),(f(1))/(1-a_((j)))}}\mathbb{P}_{M / K}=\left\{f \in \mathbb{K}:\|f\|_{L}=\max \left\{\frac{f(0)}{x_{0}}, \frac{f(1)}{1-a_{(j)}}\right\}\right\}PM/K={f∈K:‖f‖L=max{f(0)x0,f(1)1−a(j)}}
[4], Lemma 1).
In this case the cone C ⊂ Ker P M / K C ⊂ Ker P M / K C sub KerP_(M//K)C \subset \operatorname{Ker} P_{M / K}C⊂KerPM/K is given by
(15) C := { h ∈ Ker r P M / K : h ( x ) = a ( x − x 0 ) , for x ∈ [ 0 , x 0 ] (15) C := h ∈ Ker r P M / K : h ( x ) = a x − x 0 ,  for  x ∈ 0 , x 0 {:(15)C:={h inKer^(r)P_(M//K):h(x)=a(x-x_(0))," for "x in[0,x_(0)]:}:}\begin{equation*} C:=\left\{h \in \mathrm{Ker}^{r} P_{M / K}: h(x)=a\left(x-x_{0}\right), \text { for } x \in\left[0, x_{0}\right]\right. \tag{15} \end{equation*}(15)C:={h∈KerrPM/K:h(x)=a(x−x0), for x∈[0,x0]
and
h ( x ) = β ( x − x 0 ) , for x ∈ ( x 0 , 1 ] , α ⩽ 0 and β ⩾ 0 } . h ( x ) = β x − x 0 , for  x ∈ x 0 , 1 , α ⩽ 0  and  β ⩾ 0 . {:h(x)=beta(x-x_(0))", for "x in(x_(0),1],alpha <= 0" and "beta >= 0}.\left.h(x)=\beta\left(x-x_{0}\right) \text {, for } x \in\left(x_{0}, 1\right], \alpha \leqslant 0 \text { and } \beta \geqslant 0\right\} .h(x)=β(x−x0), for x∈(x0,1],α⩽0 and β⩾0}.
Then every f ∈ K f ∈ K f in Kf \in Kf∈K can be uniquely written in the form f = u f + c f f = u f + c f f=u_(f)+c_(f)f=u_{f}+c_{f}f=uf+cf, u f ∈ U u f ∈ U u_(f)in Uu_{f} \in Uuf∈U and c f ∈ C c f ∈ C c_(f)in Cc_{f} \in Ccf∈C where
(16) c f ( x ) = − f ( 0 ) x 0 ( x − x 0 ) , fol x ∈ [ 0 , x 0 ] = f ( 1 ) 1 − x 0 ( x − x 0 ) , for x ∈ ( x 0 , 1 ] (16) c f ( x ) = − f ( 0 ) x 0 x − x 0 ,  fol  x ∈ 0 , x 0 = f ( 1 ) 1 − x 0 x − x 0 ,  for  x ∈ x 0 , 1 {:[(16)c_(f)(x)=-(f(0))/(x_(0))(x-x_(0))","" fol "x in[0,x_(0)]],[=(f(1))/(1-x_(0))(x-x_(0))","" for "x in(x_(0),1]]:}\begin{align*} & c_{f}(x)=-\frac{f(0)}{x_{0}}\left(x-x_{0}\right), \text { fol } x \in\left[0, x_{0}\right] \tag{16}\\ & =\frac{f(1)}{\mathbb{1}-x_{0}}\left(x-x_{0}\right), \text { for } x \in\left(x_{0}, 1\right] \end{align*}(16)cf(x)=−f(0)x0(x−x0), fol x∈[0,x0]=f(1)1−x0(x−x0), for x∈(x0,1]
and
(17) u f ( x ) = f ( x ) + f ( 0 ) x 0 ( x − x 0 ) , u f ( x ) = f ( x ) + f ( 0 ) x 0 x − x 0 , quadu_(f)(x)=f(x)+(f(0))/(x_(0))(x-x_(0)),quad\quad u_{f}(x)=f(x)+\frac{f(0)}{x_{0}}\left(x-x_{0}\right), \quaduf(x)=f(x)+f(0)x0(x−x0), for x ∈ [ 0 , x 0 ] x ∈ 0 , x 0 quad x in[0,x_(0)]\quad x \in\left[0, x_{0}\right]x∈[0,x0]
= f ( x ) − f ( 1 ) 1 − x 0 ( x − x 0 ) , for x ∈ [ x 0 , 1 ] . = f ( x ) − f ( 1 ) 1 − x 0 x − x 0 ,  for  x ∈ x 0 , 1 . =f(x)-(f(1))/(1-x_(0))(x-x_(0))," for "x in[x_(0),1].=f(x)-\frac{f(1)}{1-x_{0}}\left(x-x_{0}\right), \text { for } x \in\left[x_{0}, 1\right] .=f(x)−f(1)1−x0(x−x0), for x∈[x0,1].
In fact
U := { u f ∈ M ∣ u f defined by (17), f ∈ K } and C := { c f ∈ K ∣ c f defined by (16), f ∈ K } . U := u f ∈ M ∣ u f  defined by (17),  f ∈ K  and  C := c f ∈ K ∣ c f  defined by (16),  f ∈ K . {:[U:={u_(f)in M∣u_(f)" defined by (17), "f in K}" and "],[quad C:={c_(f)in K∣c_(f)" defined by (16), "f in K}.]:}\begin{gathered} U:=\left\{u_{f} \in M \mid u_{f} \text { defined by (17), } f \in K\right\} \text { and } \\ \quad C:=\left\{c_{f} \in K \mid c_{f} \text { defined by (16), } f \in K\right\} . \end{gathered}U:={uf∈M∣uf defined by (17), f∈K} and C:={cf∈K∣cf defined by (16), f∈K}.
A continuous, additive and positively homogeneous selection for P M / K P M / K P_(M//K)P_{M / K}PM/K is given by q ( f ) = u f , f ∈ K q ( f ) = u f , f ∈ K q(f)=u_(f),f in Kq(f)=u_{f}, f \in Kq(f)=uf,f∈K, a fact which can be immediately verified. Remarks. Theorem A gives only a sufficient condition for the existence of an additive, positively homogeneous and continuous selection for P M / K P M / K P_(M//K)P_{M / K}PM/K.
Simple examples show that this condition is not necessary for the existence of a selection with the above-mentioned properties.
Let X = R 2 X = R 2 X=R^(2)X=\mathbb{R}^{2}X=R2 endowed with the Enclidean norm and let
and
K := { ( x , y ) ∈ R 2 : y = 2 x , x ⩾ 0 } K := ( x , y ) ∈ R 2 : y = 2 x , x ⩾ 0 K:={(x,y)inR^(2):y=2x,x >= 0}K:=\left\{(x, y) \in \mathbb{R}^{2}: y=2 x, x \geqslant 0\right\}K:={(x,y)∈R2:y=2x,x⩾0}
M := { ( x , 0 ) : x ∈ R } . M := { ( x , 0 ) : x ∈ R } . M:={(x,0):x inR}.M:=\{(x, 0): x \in \mathbb{R}\} .M:={(x,0):x∈R}.
Then, obviously, the subspace M M MMM is K K KKK-Chebyshevian and Ker P M / K = { ( 0 , 0 ) } P M / K = { ( 0 , 0 ) } P_(M//K)={(0,0)}P_{M / K}=\{(0,0)\}PM/K={(0,0)}.
In this case P m / k P m / k P_(m//k)P_{m / k}Pm/k is a continuous, positively homogeneous and additive application from K K KKK to M M MMM, but K K KKK does not admit any decomposition of the form (5).

REFERENCES

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Romanian), Ed. Acad., Bucuresti, 1967. Spaces by Elcments of Vector Subspaces (in Received & X 1993
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