Abstract

We prove that, if W⊂RnW \subset \mathbb{R}^n is a locally strongly convex body (not necessarily compact), then for any open set V⊃∂WV \supset \partial W and ε>0\varepsilon>0, there exists a C2C^2 locally strongly convex body Wε,VW_{\varepsilon, V} such that Hn−1(∂Wε,V△ ∂W)<ε\mathcal{H}^{n-1}(\partial W_{\varepsilon, V}\triangle\,\partial W)<\varepsilon and ∂Wε,V⊂V\partial W_{\varepsilon, V}\subset V. Moreover, if WW is strongly convex, then Wε,VW_{\varepsilon, V} is strongly convex as well.

Author information

Contact details are reproduced from the original publication and may be historical.

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 Hajłasz

Department of Mathematics, University of Pittsburgh, U.S.A.

hajlasz@pitt.edu

Keywords

  • Convex function
  • convex body
  • approximation
  • Lusin property

Mathematics Subject Classification

26B25; 41A29, 52A20, 52A27

References

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Suggested citation

D. Azagra, M. Drake, P. Hajłasz. “C^2-Lusin Approximation of Strongly Convex Bodies.” Journal of Convex Analysis 32 (2025), No. 1, 91–106. https://doi.org/10.68381/jca32007

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.