Abstract
Authors
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian, Romania
Keywords
Paper coordinates
C. Mustăţa, On the selections associated to the metric projection, Rev. Anal. Numér. Théor. Approx. 23 (1994) 1, 89-93 (MR # 96b: 46019)
About this paper
Journal
Revue d’Analyse Numer. Theor. Approx.
Publisher Name
Publishing Romanian Academy
Print ISSN
2501-059X
Online ISSN
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.
Paper (preprint) in HTML form
ON THE SELECTIONS ASSOCIATED TO THE METRIC PROJECTION
1. INTRODUCTION
Let XX be a real normed space and MM a closed subspace of XX. The distance from a point x in Xx \in X to MM is defined by
{:(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*}
Let
{:(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*}
be the set of elements of best approximation for xx in MM. The subspace MM is called proximinal (Chebyshevian) if P_(M)(x)!=O/P_{M}(x) \neq \varnothing (respectively P_(M)(x)P_{M}(x) is a singleton), for all x in Xx \in X. If MM is a proximinal subspace of XX, the multivalued operator P_(M):X rarr2^(M)P_{M}: X \rightarrow 2^{M} is called the metric projection of XX onto MM.
An application p:X-Mp: X-M such that p(x)inP_(M)(x)p(x) \in P_{M}(x), for all x in Xx \in X, is called a selection for P_(M)\boldsymbol{P}_{M}.
The set
{:(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*}
is called the kernel of the metric projection.
If KK is a subset; of XX and MM is proximinal (Chebyshevian) only for the elements of KK, then the subspace MM is called KK-proximinal (respectively KK-Chebyshevian).
If KK is a subset; of XX and MM is proximinal (Chebyshevian) only for the elements of KK, then the subspace MM is called KK-proximinal (respectively KK-Chebyshevian).
The restriction of the metric projection P_(M)P_{M} to KK is denoted by P_(M//M)P_{M / M}
{:(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*}
is called the kernel of the metric projection relative to KK.
For two subsets U,VU, V of XX, their sum is defined by U+V:=={u+v:u in U,v in V}U+V:= =\{u+v: u \in U, v \in V\}. If eyery x in U+Vx \in U+V can be uniquely written in the form x=u+vx=u+v, for u in Uu \in U and v in Vv \in V, then this sum is called the algebraic direct sum of the sets UU and VV and is denoted by U^(**)+VU^{*}+V. If K==U+VK= =U+V and the application (u,v)rarr u+v(u, v) \rightarrow u+v is a topological homeonorphism between U xx VU \times V and KK, then KK is called the topological direct sum of the set UU and VV and is denoted by K=U o+VK=U \oplus V.
For two subsets U,VU, V of XX, their sum is defined by U+V:=={u+v:u in U,v in V}U+V:= =\{u+v: u \in U, v \in V\}. If eyery x in U+Vx \in U+V can be uniquely written in the form x=u+vx=u+v, for u in Uu \in U and v in Vv \in V, then this sum is called the algebraic direct sum of the sets UU and VV and is denoted by U^(**)+VU^{*}+V. If K==U+VK= =U+V and the application (u,v)rarr u+v(u, v) \rightarrow u+v is a topological homeonorphism between U xx VU \times V and KK, then KK is called the topological direct sum of the set UU and VV and is denoted by K=U o+VK=U \oplus V.
MAIN THEOREM
Obviously that for K sub XK \subset X and a KK-proximinal subspace MM of XX, the properties of the selection associated to the metric projection P_(M//K)P_{M / K} depend, on one side, on the properties of the subspace MM and on the other side, on the properties of the set KK.
F. Deutsch [1] characterized the proximinal subspace MM of XX for which the metric projection P_(M)P_{M} admits a linear and continuous selection : namely, P_(M)P_{M} admits a continuous linear selection if and only if Ker P_(M)P_{M} contains a closed subspace complementary to MM (see [1], Theorem 2.2).
F. Deutsch [1] characterized the proximinal subspace MM of XX for which the metric projection P_(M)P_{M} admits a linear and continuous selection : namely, P_(M)P_{M} admits a continuous linear selection if and only if Ker P_(M)P_{M} contains a closed subspace complementary to MM (see [1], Theorem 2.2).
The aim of this paper is to answer the following question : For a closed cone KK in XX and a KK-proximinal subspace MM of XX when does the metric projection P_(M//K)P_{M / K} admit a continuous, additive and positively homogeneous selection?
A cone is a nonvoid subset KK of XX such that : aa ) x+y in Kx+y \in K, for all {:x,y inK_(i);b)lambda x in K\left.x, y \in K_{i} ; b\right) \lambda x \in K, for all lambda >= 0\lambda \geqslant 0 and x inKx \in \mathbb{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 KK be a closed cone in XX and MM a KK-proximinal subspace of XX. If the subspace MM contains a dosed cone UU and Ker P_(a//a)P_{a / a} contains a closed cone CC such that
{:(5)K^(˙)=U o+C",":}\begin{equation*}
\dot{K}=U \oplus C, \tag{5}
\end{equation*}
then P_(M//K)P_{M / K} admits a positively homogeneous, additive and continuous selection.
Proof. Let UU be a closed cone in MM and CC a closed cone in KorP_(1//pi)K o r P_{1 / \pi} such that K=U o+CK=U \oplus C.
Then, for h in Kh \in K, there exist uniquely devermined elements u_(h)in Uu_{h} \in U and c_(h)in Cc_{h} \in C such that h=u_(h)+c_(h)h=u_{h}+c_{h}.
Define the application q:K rarr MIq: K \rightarrow M I by
q(h)=u_(h),quad h inK.q(h)=u_{h}, \quad h \in \mathbb{K} .
Since KK is homeomorph to U xx OU \times O it follows that the application qq is continuous.
Let h_(1),h_(2)in Kh_{1}, h_{2} \in K and u_(h_(1)),u_(h_(2))in Uu_{h_{1}}, u_{h_{2}} \in U and c_(h_(1)),e_(h_(2))in Cc_{h_{1}}, e_{h_{2}} \in 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}}. Then
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),
showing that the application qq is additive.
Also, for lambda >= 0\lambda \geqslant 0 and h in Kh \in K it follows lambda h in K\lambda h \in K and
Also, for lambda >= 0\lambda \geqslant 0 and h in Kh \in K it follows lambda h in K\lambda h \in K and
q(lambda h)=lambdau_(th)=lambda q(h),q(\lambda h)=\lambda u_{t h}=\lambda q(h),
showing that qq is positively homogeneous.
The inclusion C subeC \subseteq Ker P_(a//K)P_{a / K} implies that for h in K,h=u_(h)+c_(h^('))h \in K, h=u_{h}+c_{h^{\prime}}, u_(h)in U,c_(h)in Cu_{h} \in U, c_{h} \in C, we have
The inclusion C subeC \subseteq Ker P_(a//K)P_{a / K} implies that for h in K,h=u_(h)+c_(h^('))h \in K, h=u_{h}+c_{h^{\prime}}, u_(h)in U,c_(h)in Cu_{h} \in U, c_{h} \in 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\left(u_{h}+c_{h}\right)=q\left(u_{h}\right)+q\left(c_{h}\right)=q\left(u_{h}\right)
and therefore
||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) .
But
d(u^(')+c_(h),M)=d(c_(h),M)d\left(u^{\prime}+c_{h}, M\right)=d\left(c_{h}, M\right)
for every u^(')in Mu^{\prime} \in M, so that
||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),
which shows that q(h)q(h) is a best approximation element for hh in MM.
In conclusion, the application q:K rarr Mq: K \rightarrow M is an additive, positively homogeneous and continuous selection of P_("M/K ")P_{\text {M/K }}
In conclusion, the application q:K rarr Mq: K \rightarrow M is an additive, positively homogeneous and continuous selection of P_("M/K ")P_{\text {M/K }}
APPLICATIONS
1^(@)1^{\circ} Let X:=Lip_(0)[0,1]X:=\operatorname{Lip}_{0}[0,1] be the linear space
(6) Lip[0,1]:={f∣f:[0,1]rarrR,f\operatorname{Lip}[0,1]:=\{f \mid f:[0,1] \rightarrow \mathbb{R}, f is Lipschitz on [0,1][0,1] and f(0)==0}f(0)= =0\}, with the Lipschitz norm
(6) Lip[0,1]:={f∣f:[0,1]rarrR,f\operatorname{Lip}[0,1]:=\{f \mid f:[0,1] \rightarrow \mathbb{R}, f is Lipschitz on [0,1][0,1] and f(0)==0}f(0)= =0\}, with the Lipschitz norm
{:(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*}
Let
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\}
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\} .
Obviously that MM is a KK-proximinal closed subspace of Lip_(0)[0,1]∼\operatorname{Lip}_{0}[0,1] \sim if f in Kf \in K then g_(0)in Mg_{0} \in M is an element of best approximation for ff if and only if g_(0)=f-Fg_{0}=f-F, where F(x)=f(1)x,x in[0,1]F(x)=f(1) x, x \in[0,1] ([4], Lemma 1).
In fact g_(0)=f-Fg_{0}=f-F, with FF given above, is a unique element of best approximation for ff in MM, which means that MM is a KK-Chebyshevian subspace of Lip_(0)[0,1]\operatorname{Lip}_{0}[0,1].
We have
||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)
and
KerP_(u//K)={f in K:||f||_(L)=f(1)}=\operatorname{Ker} P_{u / K}=\left\{f \in K:\|f\|_{L}=f(1)\right\}=
={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\}
In this case
C=Ker^(P_(u//kappa))C=\operatorname{Ker}^{P_{u / \kappa}}
and every function f in Kf \in K can be uniquely written in the form f=g+hf=g+h with g in Ug \in U and h in Ch \in C, where h(x)=f(1)x,x in[0,1]h(x)=f(1) x, x \in[0,1] and
{:[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}
Since MM is KK-Chebyshevian it follows that the metric projection operator P_(M//K)P_{M / K} is one-valued, continuous, additive and positively homogeneous.
2^(@)2^{\circ} For x_(0)in(0,1)x_{0} \in(0,1) fixed, consider the linear space
(10) X:=Lip_(x_(0))[0,1]={f//f:[0,1]rarrRX:=\operatorname{Lip}_{x_{0}}[0,1]=\{f / f:[0,1] \rightarrow \mathbb{R}, f is Lipschitz on [0,1][0,1] and {:f(x_(0))=0}\left.f\left(x_{0}\right)=0\right\}, with the Lipschits norm (7).
2^(@)2^{\circ} For x_(0)in(0,1)x_{0} \in(0,1) fixed, consider the linear space
(10) X:=Lip_(x_(0))[0,1]={f//f:[0,1]rarrRX:=\operatorname{Lip}_{x_{0}}[0,1]=\{f / f:[0,1] \rightarrow \mathbb{R}, f is Lipschitz on [0,1][0,1] and {:f(x_(0))=0}\left.f\left(x_{0}\right)=0\right\}, with the Lipschits norm (7).
Let
(11)
(11)
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\},
the annihilator in Lip_(x_(0))[0,1]\operatorname{Lip}_{x_{0}}[0,1] of the set {0,x_(0),1}\left\{0, x_{0}, 1\right\}, and
(12) 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., for all x in[0,1]x \in[0,1] ).
(12) 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., for all x in[0,1]x \in[0,1] ).
Again, MM is a KK-proximinal subspace of Lip_(x_(0))[0,1]\mathrm{Lip}_{x_{0}}[0,1].
Indeed, for f in Kf \in K let
Indeed, for f in Kf \in K let
{:[(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*}
be the set of the extensions to [0,1][0,1] of the function f|_({0,x_(0),1})\left.f\right|_{\left\{0, x_{0}, 1\right\}} which preserver the Lipschitz norm of ff on {0,x_(0),1}\left\{0, x_{0}, 1\right\}.
Then
(14)
and
(14)
and
P_(M//K)(f)=f-E(f)P_{M / K}(f)=f-E(f)
Ker 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\}
[4], Lemma 1).
In this case the cone C sub KerP_(M//K)C \subset \operatorname{Ker} P_{M / K} is given by
[4], Lemma 1).
In this case the cone C sub KerP_(M//K)C \subset \operatorname{Ker} P_{M / K} is given by
{:(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*}
and
{: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\} .
Then every f in Kf \in K can be uniquely written in the form f=u_(f)+c_(f)f=u_{f}+c_{f}, u_(f)in Uu_{f} \in U and c_(f)in Cc_{f} \in C where
{:[(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*}𝟙
and
(17) 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), \quad for quad x in[0,x_(0)]\quad x \in\left[0, x_{0}\right]
(17) 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), \quad for quad x in[0,x_(0)]\quad x \in\left[0, x_{0}\right]
=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] .
In fact
{:[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}
A continuous, additive and positively homogeneous selection for P_(M//K)P_{M / K} is given by q(f)=u_(f),f in Kq(f)=u_{f}, f \in 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}.
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=\mathbb{R}^{2} endowed with the Enclidean norm and let
and
and
K:={(x,y)inR^(2):y=2x,x >= 0}K:=\left\{(x, y) \in \mathbb{R}^{2}: y=2 x, x \geqslant 0\right\}
M:={(x,0):x inR}.M:=\{(x, 0): x \in \mathbb{R}\} .
Then, obviously, the subspace MM is KK-Chebyshevian and Ker P_(M//K)={(0,0)}P_{M / K}=\{(0,0)\}.
In this case P_(m//k)P_{m / k} is a continuous, positively homogeneous and additive application from KK to MM, but KK does not admit any decomposition of the form (5).
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- Dentsch, F., Linear Selections for the Metric Projocion, J. Funct. Aralysis 49 (1982) 260-292.
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- Muths, Revue d'Anal Nowed the Me Shone's Extension Thcorem for Lipschile Func-195-145.
(1977) 222-230. Extension of Lipschils Functions, J.A.T. Ig Mustăla, C
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