Abstract
Let f be a continuous real function on a convex subset of a Banach space. We study what can be said about the semiconcavity (with a general modulus) of f, if we know that the estimate.
Delta(2)(h)(f; x) [0; infinity) is a nondecreasing function right continuous at 0 with omega(0) = 0. A partial answer to this question was given by P.
Cannarsa and C. Sinestrari (2004); we prove versions of their result, which are in a sense best possible.
We essentially use methods of A. Marchaud, S.
B. Stechkin and others, whose results clarify when the inequality vertical bar Delta(2)(h)(f; x)vertical bar <=omega (parallel to h parallel to) implies that f is a C-1 function (and f ' is uniformly continuous with a corresponding modulus of continuity).