A convexity criterion via the De Giorgi slope

Tahar Z. Boulmezaoud, Aris Daniilidis, Trí Minh Lê

Let $X$ be a Banach space and $f\in {\mathcal{C}}^1(X)$ be bounded from below. We show that if for some $m\ge 1$, the function $x\mapsto \Vert \nabla f(x){\Vert }^m$ is convex, then $f$ is con…