Journal of Convex Analysis 32 (2025), No. 1, 091--106
Copyright Heldermann Verlag 2025
C2-Lusin Approximation of Strongly Convex Bodies
Daniel Azagra
Dept. of Math. Analysis and Applied Mathematics, Universidad Complutense, Madrid, Spain
azagra@mat.ucm.es
Marjorie Drake
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, U.S.A.
mkdrake@mit.edu
Piotr Hajlasz
Department of Mathematics, University of Pittsburgh, U.S.A.
hajlasz@pitt.edu
[Abstract-pdf]
We prove that, if $W \subset \mathbb{R}^n$ is a locally strongly convex body (not necessarily compact), then for any open set $V \supset \partial W$ and $\varepsilon>0$, there exists a $C^2$ locally strongly convex body $W_{\varepsilon, V}$ such that $\mathcal{H}^{n-1}(\partial W_{\varepsilon, V}\triangle\,\partial W)<\varepsilon$ and $\partial W_{\varepsilon, V}\subset V$. Moreover, if $W$ is strongly convex, then $W_{\varepsilon, V}$ is strongly convex as well.
Keywords: Convex function, convex body, approximation, Lusin property.
MSC: 26B25; 41A29, 52A20, 52A27.
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