Lipschitz selections of convexifications of pseudo-lipschitz multifunctions and the Lipschitz maximum principle for differential inclusions

We prove that, if X, Y are finite-dimensional real linear spaces and F : X → 2^Y is a multifunction that has the pseudo-Lipschitz property at a point (x₀, y₀) ∈ Graph(F), then for every ε >; 0 there exists a Lipschitz multifunction V^ε : N(ε) → 2^Y , defined on a neighborhood N(ε) of x₀, such that (i) V^ε has compact convex values, (ii) V^ε(x₀) = {y₀}, and (iii) for every x ∈ N(ε), V^ε(x) is a subset of the convex hull co(F^ε(x)) of the intersection F^ε(x) of F(x) with the closed ε-ball centered at y₀. In particular, this implies the existence of a Lipschitz single-valued selection f^ε of co(F^ε) near x₀ satisfying f^ε(x₀) = y₀.