The Monotonicity of the Cheeger constant for Parallel Bodies

Ftouhi I (2025)


Publication Type: Journal article

Publication year: 2025

Journal

Book Volume: 206

Article Number: 44

Journal Issue: 2

DOI: 10.1007/s10957-025-02727-z

Abstract

We prove that for every planar convex set Ω, the function t∈(-r(Ω),+∞)⟼|Ωt|h(Ωt) is monotonically decreasing, where r, |·| and h stand for the inradius, the measure and the Cheeger constant and (Ωt) for parallel bodies of Ω. The result is shown not to hold when the convexity assumption is dropped. We also prove the differentiability of the map t⟼h(Ωt) in any dimension and without any regularity assumption on the convex Ω, obtaining an explicit formula for the derivative. Those results are then combined to obtain estimates on the contact surface of the Cheeger sets of convex bodies. Finally, potential generalizations to other functionals such as the first eigenvalue of the Dirichlet Laplacian are explored.

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How to cite

APA:

Ftouhi, I. (2025). The Monotonicity of the Cheeger constant for Parallel Bodies. Journal of Optimization Theory and Applications, 206(2). https://doi.org/10.1007/s10957-025-02727-z

MLA:

Ftouhi, Ilias. "The Monotonicity of the Cheeger constant for Parallel Bodies." Journal of Optimization Theory and Applications 206.2 (2025).

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