## 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 $$x_{1},x_{2}\in T_{R}$$.

### MSC:

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

 [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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