On functions of nonconvexity for graphs of continuous functions (Q1910110)

For a subset \(P\) of a normed linear space, the ``degree'' of non-convexity of the set \(P\) is introduced as a function \(h_P:(0,\infty)\to [0,2]\). It turns out that for a closed \(P\), the convexity of \(P\) is equivalent to \(h_P\equiv 0\). A special attention is given to the case when \(P\) is a subset of \(\mathbb{R}^2\) or a graph of a continuous function of one variable. The main application of functions \(h_P\) in the paper is that in the Michael Selection Theorem the condition of convexity can be replaced by the condition that the values of a multivalued map are graphs of polynomials \(g(x)= x^n+ a_{n-1}x^{n-1}+\cdots+ a_1x+ a_0\), \(|a_i|\leq C\).