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 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