
Journal of Convex Analysis 25 (2018), No. 1, 241269 Copyright Heldermann Verlag 2018 On Semiconcavity via the Second Difference Ludek Zajícek Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic zajicek@karlin.mff.cuni.cz [Abstractpdf] 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). Keywords: Semiconcave function with general modulus, second difference, second modulus of continuity, Jensen semiconcave function, alphamidconvex function, semiZygmund class. MSC: 26B25; 46T99 [ Fulltextpdf (184 KB)] for subscribers only. 