A convexity criterion via the De Giorgi slope

Abstract

Let X be a Banach space and f ∈ C1(X) be bounded from below. We show that if for some m ≥1,thefunction x → ∇f(x)m is convex, then f is convex. We also establish a more general version of this result: if f is continuous and bounded from below, then it is convex, provided x → sf(x)m is convex for some m ≥ 1, where sf denotes the (De Giorgi) metric slope of f.