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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:
The norm of \(X^*\) is strongly sub-differentiable and, for every \(f\in S_{X^*}\), the set \(J_X(f)\) is compact.
\(X\) is reflexive and the relative weak and the norm topologies coincide on the unit sphere \(S_X\) of \(X\).
Every closed convex subset of \(X\) is approximatively compact.
Every closed convex subset of \(X\) is strongly proximinal.

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Full Text: DOI
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