Local properties of geodesics on \(p\)-convex sets
Let \(M\) be a set of points of a Hilbert space \(H\) and \(p: M\to\mathbb{R}^ +\) be continuous. \(M\) is said to be a \(p\)-convex set iff \((a,v-u)\leq p(u)| a| | v-u|^ 2\) whenever \(u,v\in M\) and \(a\in \partial^ - I_ M(u)\), where \(\partial^ - I_ M(u)\) denotes a closed convex cone to \(M\) at \(u\) in \(H\). After proving some local properties of geodesics on \(p\)-convex sets the author gives an intrinsic characterization of geodesics in terms of local minimality for the energy functional, proves local uniqueness and continuous dependence on extrema for geodesics with assigned endpoints, and shows some consequences of these results.
scientific article; zbMATH DE number 59763
scientific article