ON FUNCTIONS OF NONCONVEXITY FOR GRAPHS OF CONTINUOUS-FUNCTIONS

For any subset P of a normed space we introduce the concept of a funct ion h(p) of non-convexity of the set P. We investigate the case when P lies in the Euclidean plane and P is the graph of some continuous fun ction of one variable. One of the applications for example is that in the well-known E. Michael Selection Theorem the condition of convexity in this case can be replaced by the condition that the values of the many-valued map are graphs of polynomials g(x) = x(n) + a(n-1)x(n-1) ...+ a(1)x + a(0) \ai\ less than or equal to C. Here, the coordinate system is not fixed: it may be different for different values of the m any-valued map. (C) 1995 Academic Press, Inc.