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Journal of Convex Analysis 27 (2020), No. 1, 079--102
Copyright Heldermann Verlag 2020

Prescribing Tangent Hyperplanes to C1,1 and C1, ω Convex Hypersurfaces in Hilbert and Superreflexive Banach Spaces

Daniel Azagra
Dep. de Análisis Matemático y Matemática Aplicada, Facultad Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain

Carlos Mudarra
Instituto de Ciencias Matemáticas, 28049 Madrid, Spain


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 $1+\alpha$, $\alpha\in (0, 1]$.

Keywords: Convex body, convex function, Whitney extension theorem, differentiability, signed distance function.

MSC: 6B05, 26B25, 52A05, 52A20.

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