Prescribing Tangent Hyperplanes to C^1,1 and C^{1, ω} Convex Hypersurfaces in Hilbert and Superreflexive Banach Spaces

D. Azagra ; C. Mudarra

Journal of convex analysis, Tome 27 (2020) no. 1, pp. 79-102

Résumé

Let $X$ denote $\mathbb{R}^n$ or, more generally, a Hilbert space. Given an arbitrary subset $C$ of $X$ and a collection $\mathcal{H}$ of affine hyperplanes of $X$ such that every $H\in\mathcal{H}$ passes through some point $x_{H}\in C$, and $C=\{x_H : H\in\mathcal{H}\}$, what conditions are necessary and sufficient for the existence of a $C^{1,1}$ convex hypersurface $S$ in $X$ such that $H$ is tangent to $S$ at $x_H$ for every $H\in\mathcal{H}$? In this paper we give an answer to this question. We also provide solutions to similar problems for convex hypersurfaces of class $C^{1, \omega}$ in Hilbert spaces, and for convex hypersurfaces of class $C^{1, \alpha}$ in superreflexive Banach spaces having equivalent norms with moduli of smoothness of power type $1+\alpha$, $\alpha\in (0, 1]$.

Détail

Classification : 6B05, 26B25, 52A05, 52A20
Mots-clés : Convex body, convex function, Whitney extension theorem, differentiability, signed distance function