Geodesic
Two Conditions for a Function to be Convex
Journal of convex analysis, Tome 20 (2013) no. 4, pp. 1189-1201.

Voir la notice de l'article provenant de la source Heldermann Verlag

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. 
@article{JCA_2013_20_4_JCA_2013_20_4_a16,
     author = {A. O. Caruso and A. Villani},
     title = {Two {Conditions} for a {Function} to be {Convex}},
     journal = {Journal of convex analysis},
     pages = {1189--1201},
     publisher = {mathdoc},
     volume = {20},
     number = {4},
     year = {2013},
     url = {http://geodesic.mathdoc.fr/item/JCA_2013_20_4_JCA_2013_20_4_a16/}
}
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A. O. Caruso; A. Villani. Two Conditions for a Function to be Convex. Journal of convex analysis, Tome 20 (2013) no. 4, pp. 1189-1201. http://geodesic.mathdoc.fr/item/JCA_2013_20_4_JCA_2013_20_4_a16/