Continuous selections of multifunctions with weakly convex values
The author shows that given a paracompact space \(X\), numbers \(0<\varepsilon<R\) and a lower semicontinuous set-valued map \(F\) defined on \(X\) with closed values in a Hilbert space \(H\) such that for any \(x\in X\), \(F(x)\) is weakly convex with respect to \(R\), \(F\) admits a continuous selection provided it has an \(\varepsilon\)-selection. A set \(A\subset H\) is weakly convex if (roughly speaking) for any \(x_0,x_1\in A\), the segment \([x_0,x_1]\) meets \(A\) at a point \(x\) different from \(x_0\) and \(x_1\). In a similar spirit the author shows the existence of a continuous selection of a lower semicontinuous map \(F\) defined on a paracompact space \(X\) of dimension \(n+1\), provided the values of \(F\) are \(n\)-connected, closed and weakly convex with respect to \(R>0\).
scientific article; zbMATH DE number 5275707
scientific article