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Reverse Faber–Krahn and Mahler inequalities for the Cheeger constant

Part of: Manifolds General convexity

Published online by Cambridge University Press:  22 June 2018

Dorin Bucur  and
Ilaria Fragalà
Dorin Bucur
Affiliation:
Laboratoire de Mathématiques, UMR 5127 and Institut Universitaire de France, Université Savoie Mont Blanc, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France (dorin.bucur@univ-savoie.fr)
Ilaria Fragalà
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy (ilaria.fragala@polimi.it)
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Abstract

We prove a reverse Faber–Krahn inequality for the Cheeger constant, stating that every convex body in ℝ2 has an affine image such that the product between its Cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle. An analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square. We also prove a Mahler-type inequality for the Cheeger constant, stating that every axisymmetric convex body in ℝ2 has a linear image such that the product between its Cheeger constant and the Cheeger constant of its polar body is not larger than the same quantity for the square.

Keywords

Cheeger constantshape optimizationspectral inequalitiesconvex bodies

MSC classification

Primary: 49Q10: Optimization of shapes other than minimal surfaces 52A40: Inequalities and extremum problems 52A10: Convex sets in $2$ dimensions (including convex curves)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 5 , October 2018 , pp. 913 - 937
Copyright
Copyright © Royal Society of Edinburgh 2018 

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