On Semiconcavity via the Second Difference

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 $\Delta_h^2(f,x) \leq \omega(\|h\|)$ holds, where $\Delta_h^2(f,x) = f(x+2h)-2f(x+h) + f(x)$ and $\omega:[0,\infty) \to [0,\infty)$ 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 $|\Delta_h^2(f,x)| \leq \omega(\|h\|)$ implies that $f$ is a $C^1$ function (and $f'$ is uniformly continuous with a corresponding modulus of continuity).