Two Conditions for a Function to be Convex

We present two sufficient conditions in order that a real function on a finite dimensional normed space be convex (Theorems 1.1 and 1.2) 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.