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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)
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