Paraconvexity of the graphs of Lipschitzian functions
Following \textit{E. Michael} [Math. Scand. 7, 372-376 (1960; Zbl 0093.12001)] a closed subset \(P\) of a Banach space \(B\) is called \(\alpha\)-paraconvex if for \(x\in B\), \(r> \text{dist} (x,P)\) and \(y\in\text{conv} (P\cap K(x,r))\) we have \(\text{dist} (y,P)\leq \alpha \cdot r\), where \(K(x,r): =\{z\in B:|z-x |\leq r\}\). Each lsc multifunction defined on a paracompact space \(T\) and having \(\alpha\)-paraconvex values \(F(t) \subset B\) admits a continuous selection \(f:T\to B\) if \(0\leq \alpha<1\). In the paper under review it is proved, by using direct geometrical considerations, that if \(B=X \oplus Y\), where \(x=\mathbb{R}^n\), \(Y=\mathbb{R}\), then the graphs of \(k\)-Lipschitzian functions \(g:X\to Y\) with the same constant \(k\) are \(\alpha \)-paraconvex and an explicit formula for \(\alpha\) is given. The paper contains four questions concerning the optimisation of the values of \(\alpha\) and the possibility of replacing \(X\) and \(Y\) by arbitrary Hilbert spaces, or \(Y\) by \(\mathbb{R}^n\). Reviewers notes: 1. A. Maliszewski in a private communication to the reviewer indicates the following counterexample: Let \(f:\mathbb{R} \to \mathbb{R}^2\), with the sup norm, be such that \(f(-3)= (-3,3)\), \(f(-1)= (-3,-3)\), \(f(1)= (3,-3)\), \(f(3)= (3,3)\), \(f\) periodic with the period 8 and linear in the segments \([-3,-1]\), \([-1,1]\), \([1,3]\) and \([3,5]\). Then \(f\) is Lipschitzian with \(k=3\), but the graph of \(f\) is not \(\alpha\)-paraconvex. Nevertheless a multifunction \(F:\mathbb{R}^n \to\mathbb{R}^3\), whose values are graphs of \(f\in\text{Lip}_3 (\mathbb{R}, \mathbb{R}^2)\) has continuous selection. 2. The paper contains some misprints: \(e_1, \dots, e_n\) cannot create a base for \((n+1)\)-dimensional \(V_{n+1}\) on p. 239 and for the set \(W\) on the same page the assumption of closedness is necessary.
scientific article; zbMATH DE number 955861