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Journal of Convex Analysis 22 (2015), No. 4, 1207--1214
Copyright Heldermann Verlag 2015

A Note on the Stability of the Cheeger Constant of N-gons

Marco Caroccia
Dip. di Matematica, UniversitÓ di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Robin Neumayer
Dept. of Mathematics, University of Texas, Austin, TX 78712-0257, U.S.A.

The regular N-gon provides the minimal Cheeger constant in the class of all N-gons with fixed volume. This result is due to a work of D. Bucur and I. FragalÓ [A Faber-Krahn inequality for the Cheeger Constant of N-gons, J. of Geometric Analysis (2014)]. In this note, we address the stability of their result in terms of the L1 distance between sets. Furthermore, we provide a stability inequality in terms of the Hausdorff distance between the boundaries of sets in the class of polygons having uniformly bounded diameter. Finally, we show that our results are sharp, both in the exponent of decay and in the notion of distance between sets.

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