On farthest points of sets. (English) Zbl 1124.46007

Math. Notes 80, No. 2, 159-166 (2006); translation from Mat. Zametki 80, No. 2, 163-170 (2006).
Let \(E\) be a Banach space and \(A\subset E\) be a bounded closed convex set. Let \(f_{A}:E\rightarrow\mathbb{R},\;f_{A}(x)=\sup\{\left\| x-a\right\| :a\in A\}\) be the farthest point function for \(A.\) For \(x\in E\), let \(a(x)\in A\) be the farthest point from \(x,\) i.e., \(\left\| x-a(x)\right\| =f_{A}(x)\) and \(\left\| x-a\right\| <f_{A}(x),\) for all \(a\in A\backslash\{a(x)\}.\) The set \(A\) is called strongly convex of radius \(R>0\) if it is an intersection of balls of radius \(R,\) and is called a generating set if, for every intersection of translates \(B=\bigcap_{x\in E}(A+x)\neq\emptyset\), there is a set \(A_{1}\) such that \(B+A_{1}=A.\) For \(A\subset E\) and \(r>0\), one defines \(T_{r}=T_{r}(A)=\{x\in A| \;f_{A} (x)>r\}.\) The set \(A\) is called a strong \(r\)-sun if any point \(a(x)\) that is the farthest point from a point \(x\in T_{r}\) is also the farthest point from any \(w\) lying on the segment \([x_{i}a(x)]\) and satisfying the inequality \(\left\| w-a(x)\right\| >r.\)
The authors obtain some results concerning the existence and uniqueness of farthest points, and necessary and sufficient conditions for the strong convexity of a set, such as: Theorem 1. If \(E\) is a reflexive Banach space with strongly convex and generating ball, then for every \(A\subset E,\) strongly convex of radius \(r>0,\) there exists the farthest point \(a(x),\) for every \(x\in T_{r}(A)\). Theorem 2. If \(A\) is strongly convex of radius \(r>0\) in a Hilbert space \(H,\) then for every \(x\in T_{r}(A)\) there is \(a(x)\in A\) and \(\left\| a(x_{1} )-a(x_{2})\right\| \leq\frac{r}{R-r}\left\| x_{1}-x_{2}\right\| ,\) for every \(R>r\) and \(x_{1},x_{2}\in T_{R}\).


46B20 Geometry and structure of normed linear spaces
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Full Text: DOI


[1] E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis, Fizmatlit, Moscow, 2004. · Zbl 1181.26028
[2] J. Diestel, Geometry of Banach Spaces, Lecture Notes in Math, 485, Springer, Berlin, 1975; Russian transl.: Vishcha Shkola, Kiev, 1980. · Zbl 0307.46009
[3] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984; Russian transl.: Mir, Moscow, 1988. · Zbl 0641.47066
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