Existence of a Lipschitzian selection (Q2718145)

Let \(R^{m}\) be an Euclidean space, \(\langle\cdot,\cdot\rangle\) be the inner product, \(\|\cdot\|= \sqrt{\langle\cdot,\cdot\rangle}, U(a,r)=\{x\in R^{m}:\|x-a\|\leq r\}\) and \(D\) be a compact set of \(R^{k}.\) The next theorem is the main result of the paper. NEWLINENEWLINENEWLINETheorem. Let i) \(F(x)= U(a(x),r(x)) \neq\emptyset,\) where \(a:D\to R^{m}\) and \(r:D\to R_{+}\) are Lipschitz on \(D\); ii) \(F_{0}(x)=\{f\in F(x):\langle f,n(x)\rangle \leq 0\},\) where \(n:D\to R^m\) is Lipschitz on \(D.\) Then the map \(f:D\to R^{m}\) is a Lipschitz selector of \(F\) on \(D,\) where NEWLINE\[NEWLINE f(x)= \int_{F_{0}(x)}y dy \Biggl/\int_{F_{0}(x)}dy.NEWLINE\]