Journal of Convex Analysis 20 (2013), No. 4, 1189--1201
Copyright 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.
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