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A remark on a conjecture on the symmetric Gaussian problem

Part of: Distribution theory - Probability Manifolds

Published online by Cambridge University Press:  15 April 2024

Nicola Fusco Open the ORCID record for Nicola Fusco [Opens in a new window]  and
Domenico Angelo La Manna Open the ORCID record for Domenico Angelo La Manna [Opens in a new window]
Nicola Fusco
Affiliation:
Dipartimento di Matematica ed Applicazioni R. Caccioppoli, Università degli studi di Napoli Federico II, Napoli, Campania, Italy
Domenico Angelo La Manna*
Affiliation:
Dipartimento di Matematica ed Applicazioni R. Caccioppoli, Università degli studi di Napoli Federico II, Napoli, Campania, Italy
*
Corresponding author: Domenico Angelo La Manna, email: domenicoangelo.lamanna@unina.it
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Abstract

In this paper, we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centred at the origin is the only minimizer of such a functional for certain values of the mass. We prove that this is the case in dimension 2 while in higher dimension the situation is different. In fact, for small values of mass, the ball centred at the origin is a local minimizer, while for larger values the ball is a maximizer among convex sets with a uniform bound on the curvature.

Keywords

Gauss spaceisoperimetric inequality

MSC classification

Primary: 49Q20: Variational problems in a geometric measure-theoretic setting 60E15: Inequalities; stochastic orderings

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 2 , May 2024 , pp. 643 - 661
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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