On Functions of Nonconvexity for Graphs of Continuous Functions

For any subset P of a normed space we introduce the concept of a function hP 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 function 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) = xn + an−1xn−1 + ••• + a1x + a0 |ai| ≤ C. Here, the coordinate system is not fixed: it may be different for different values of the many-valued map.