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 (x
₀
, y
₀
) ∈ 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 x
₀
, such that (i) V
^ε
has compact convex values, (ii) V
^ε
(x
₀
) = {y
₀
}, 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 y
₀
. In particular, this implies the existence of a Lipschitz single-valued selection f
^ε
of co(F
^ε
) near x
₀
satisfying f
^ε
(x
₀
) = y
₀
.