Geodesic
There are Many Totally Convex Functions
Journal of convex analysis, Tome 13 (2006) no. 3, pp. 623-632.

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

Let $K$ be a convex subset of a normed linear space and let $R^1$ denote the real line. We show that there are many (in the sense of Baire category) strictly convex and totally convex functions $f \colon K \to R^1$. It is known that the existence of such functions is crucial in numerous optimization algorithms.
Classification : 46N10, 52A41, 54E50, 54E52
Mots-clés : Complete metric space, essentially strictly convex function, generic property, strictly convex function, totally convex function 
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     title = {There are {Many} {Totally} {Convex} {Functions}},
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D. Butnariu; S. Reich; A. J. Zaslavski. There are Many Totally Convex Functions. Journal of convex analysis, Tome 13 (2006) no. 3, pp. 623-632. http://geodesic.mathdoc.fr/item/JCA_2006_13_3_JCA_2006_13_3_a8/