Increasing paths leading to a face of a convex compact set in a Hilbert space (Q1123405)

This paper is a sequel of a work of the same author in Pac. J. Math. 124, 289-294 (1986; Zbl 0593.52001), where the author proves that if the face \(F=\{x\in C:\) \(\ell (x)=\max_{y\in C} \ell (y)\}\), C being the compact set in the normed linear space E, is of infinite dimension, then for every \(n\in {\mathbb{N}}^*\), there are n-strictly increasing paths on the one-skeleton of C mutually disjoint that lead in F. Here the author proves that if the dimension of F is k, then there are \(k+1\) such paths, for every \(k\in {\mathbb{N}}^*\), and gives an example showing that this result is the best possible in a Hilbert space.