CHARACTERIZATION OF THE CLARKE REGULARITY OF SUBANALYTIC SETS

In this note, we will show that for a closed subanalytic subset A subset of R-n, the Clarke tangential regularity of A at x(0) is an element of A is equivalent to the coincidence of the Clarke tangent cone to A at x(0) with the set L(A, x(0)) := {(c) over dot(+)(0) is an element of R-n : c : [0, 1] -> A is Lipschitz, c(0) = x(0)}, where (c) over dot(+) (0) denotes the right-strict derivative of c at 0. The results obtained are used to show that the Clarke regularity of the epigraph of a function may be characterized by a new formula of the Clarke subdifferential of that function.