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Inner and outer smooth approximation of convex hypersurfaces. When is it possible? / Azagra, Daniel; Столяров, Дмитрий Михайлович.

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 230, 113225, 01.05.2023.

Research output: Contribution to journal › Article › peer-review

Harvard

Azagra, D & Столяров, ДМ 2023, 'Inner and outer smooth approximation of convex hypersurfaces. When is it possible?', Nonlinear Analysis, Theory, Methods and Applications, vol. 230, 113225. https://doi.org/10.1016/j.na.2023.113225

APA

Azagra, D., & Столяров, Д. М. (2023). Inner and outer smooth approximation of convex hypersurfaces. When is it possible? Nonlinear Analysis, Theory, Methods and Applications, 230, [113225]. https://doi.org/10.1016/j.na.2023.113225

Vancouver

Azagra D, Столяров ДМ. Inner and outer smooth approximation of convex hypersurfaces. When is it possible? Nonlinear Analysis, Theory, Methods and Applications. 2023 May 1;230. 113225. https://doi.org/10.1016/j.na.2023.113225

Author

Azagra, Daniel ; Столяров, Дмитрий Михайлович. / Inner and outer smooth approximation of convex hypersurfaces. When is it possible?. In: Nonlinear Analysis, Theory, Methods and Applications. 2023 ; Vol. 230.

BibTeX

@article{e463a9f8ede644a68e855f1bc34feba7,
title = "Inner and outer smooth approximation of convex hypersurfaces. When is it possible?",
abstract = "Let S be a convex hypersurface (the boundary of a closed convex set V with nonempty interior) in Rn. We prove that S contains no lines if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∩int(V). We also show that S contains no rays if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∖V. Moreover, in both cases, SU can be taken strongly convex. We also establish similar results for convex functions defined on open convex subsets of Rn, completely characterizing the class of convex functions that can be approximated in the C0-fine topology by smooth convex functions from above or from below. We also provide similar results for C1-fine approximations.",
keywords = "Convex body, Convex hypersurface, Fine approximation",
author = "Daniel Azagra and Столяров, {Дмитрий Михайлович}",
year = "2023",
month = may,
day = "1",
doi = "10.1016/j.na.2023.113225",
language = "English",
volume = "230",
journal = "Nonlinear Analysis, Theory, Methods and Applications",
issn = "0362-546X",
publisher = "Elsevier",

}

RIS

TY - JOUR

T1 - Inner and outer smooth approximation of convex hypersurfaces. When is it possible?

AU - Azagra, Daniel

AU - Столяров, Дмитрий Михайлович

PY - 2023/5/1

Y1 - 2023/5/1

N2 - Let S be a convex hypersurface (the boundary of a closed convex set V with nonempty interior) in Rn. We prove that S contains no lines if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∩int(V). We also show that S contains no rays if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∖V. Moreover, in both cases, SU can be taken strongly convex. We also establish similar results for convex functions defined on open convex subsets of Rn, completely characterizing the class of convex functions that can be approximated in the C0-fine topology by smooth convex functions from above or from below. We also provide similar results for C1-fine approximations.

AB - Let S be a convex hypersurface (the boundary of a closed convex set V with nonempty interior) in Rn. We prove that S contains no lines if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∩int(V). We also show that S contains no rays if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∖V. Moreover, in both cases, SU can be taken strongly convex. We also establish similar results for convex functions defined on open convex subsets of Rn, completely characterizing the class of convex functions that can be approximated in the C0-fine topology by smooth convex functions from above or from below. We also provide similar results for C1-fine approximations.

KW - Convex body

KW - Convex hypersurface

KW - Fine approximation

UR - https://www.mendeley.com/catalogue/cc4d1119-3888-3fe8-b435-b918df1d9881/

U2 - 10.1016/j.na.2023.113225

DO - 10.1016/j.na.2023.113225

M3 - Article

VL - 230

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

M1 - 113225

ER -

ID: 102518688