On paraconvexity of graphs of continuous functions

The concept of paraconvexity of a subsetP E of a normed spaceE was first introduced by E. Michael. Roughly speaking, it consists of a controlled weakening of the convexity assumption forP, where the control is guaranteed via some parameter [0, 1). In this paper, we consider the case whenP is a subset of some (n+1)-dimensional Euclidean spaceE andP is the graph of some continuous functionf:V , whereV E is some convexn-dimensional subset ofE. Our key result is that paraconvexity of such a setP follows from the paraconvexity of sections ofP by two-dimensional planes, orthogonal toV. As an application, we prove a selection theorem for graph-valued mappings whose values have Lipschitzian (with a fixed constant) or monotone two-dimensional sections.Supported in part by the Ministry of Science and Technology of the Republic of Slovenia Research Grant No. P1-0214-101-93.Supported in part by G. Soros International Science Foundation.