Abstract
In this note we obtain some properties of the Cheeger set C_varOmega associated to a k-rotationally symmetric planar convex body varOmega . More precisely, we prove that C_varOmega is also k-rotationally symmetric and that the boundary of C_varOmega touches all the edges of varOmega .
Highlights
The Cheeger problem is a classical problem in Geometry widely studied in literature, with connections in many different fields
Where the infimum in (2) is taken over all non-empty finite-perimeter sets X contained in Ω, and P (X) and V (X) denote the perimeter and the volume of X, respectively. This constant h(Ω) is usually called the Cheeger constant of Ω, and any subset X contained in Ω providing the infimum in (2) is called a Cheeger set of Ω
Vol 77 (2022) Cheeger Sets for Rotationally Symmetric Planar Convex Bodies Page 3 of 15 9 interesting characterizations of the Cheeger sets for planar convex sets. One of these results establishes a condition on a planar convex set Ω for assuring that the Cheeger set of Ω is Ω itself, in terms of an inequality involving the curvature of the boundary of Ω [27, Th. 2]
Summary
The Cheeger problem is a classical problem in Geometry widely studied in literature, with connections in many different fields. The interested reader may find in [50] some historical remarks on closely related questions The origin of this problem can be traced back to a paper by J. Cheeger [17], who proved in 1969 the following inequality for any bounded domain Ω in Rn (this result was stated for any compact Riemannian manifold without boundary): λ1(Ω).
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