Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 20 (2013), No. 4, 1189--1201Copyright Heldermann Verlag 2013 Two Conditions for a Function to be Convex Andrea Orazio Caruso Dip. di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy aocaruso@dmi.unict.it Alfonso Villani Dip. di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy villani@dmi.unict.it [Abstract-pdf] We present two sufficient conditions in order that a real function on a finite-dimensional normed space be convex (Theorems 1 and 2) and show some consequences of them. In particular, it comes out that a real function $f$ on a finite-dimensional Hilbert space $X$ is convex, provided that $f$ has the property that for each point $y \in X$ and each $\lambda > 0$ the real function $X \ni x \to \lambda f(x) + \|x-y\|^2$ has a unique global minimum. [ Fulltext-pdf  (135  KB)] for subscribers only.