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Strong proximinality of closed convex sets. (English) Zbl 1223.41023
Let $$X$$ be a Banach space with unit ball $$B_X$$, unit sphere $$S_X$$ and dual space $$X^*$$. For $$f\in X^*$$, with $$\|f\|=1$$, set $$J_X(f)=\{ x\in S_X: f(x)=1\}$$. The functional $$f$$ is norm attaining if $$J_X(f)\neq \emptyset$$.
A closed set $$C$$ is strongly proximinal if, for each $$x\in X$$ and $$\varepsilon>0$$, there exists $$t>0$$ such that $$P_C(x,t)\subset P_C(x)+\varepsilon B_X$$, where $$P_C(x)=\{ y\in C: \|x-y\|= d(x,C)\}$$ and $$P_C(x,t)=\{ y\in C: \|x-y\|< d(x,C)+t\}$$.
The set $$C$$ is approximatively compact if every minimizing sequence in $$C$$ has a convergent subsequence.
The norm $$\|\cdot\|$$ is said to be strongly sub-differentiable (SSD), if for every $$x\in S_X$$ the limit
$\lim_{t\to 0^+}\frac{\|x+th\|-\|x\|}{t}$
exists uniformly for $$h\in S_X$$.
The main result of the paper is the following:
Theorem. For a Banach space $$X$$ the following assertions are equivalent:
(i)
The norm of $$X^*$$ is strongly sub-differentiable and, for every $$f\in S_{X^*}$$, the set $$J_X(f)$$ is compact.
(ii)
$$X$$ is reflexive and the relative weak and the norm topologies coincide on the unit sphere $$S_X$$ of $$X$$.
(iii)
Every closed convex subset of $$X$$ is approximatively compact.
(iv)
Every closed convex subset of $$X$$ is strongly proximinal.

##### MSC:
 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
##### Keywords:
Strong proximinality; metric projection
Full Text:
##### References:
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