Cheeger Sets for Rotationally Symmetric Planar Convex Bodies

Abstract

In this note we obtain some properties of the Cheeger set $$C_\varOmega $$ C Ω associated to a k-rotationally symmetric planar convex body $$\varOmega $$ Ω . More precisely, we prove that $$C_\varOmega $$ C Ω is also k-rotationally symmetric and that the boundary of $$C_\varOmega $$ C Ω touches all the edges of $$\varOmega $$ Ω .

The Cheeger problem in abstract measure spaces

“…In particular, (P.6) holds with $$f(\varepsilon)=n\omega _n^{1/n}\varepsilon ^{-1/n}$$, and follows from the standard isoperimetric inequality $$$$\begin{equation} P_{\rm Eucl}(E)\geqslant n\omega _n^{1/n} \mathcal L^n(E)^{1-\frac{1}{n}}, \end{equation}$$$$holding for any $$E\in \mathcal A$$. The Cheeger problem in this setting is standard, see [91, 112], and its minimizers are now completely characterized for a large class of planar sets [41, 88, 92, 95, 114], and reasonably well‐understood for convex $$N$$‐dimensional bodies [5, 26]. Recently, Cheeger clusters have been introduced and studied in [47], see also [30, 31, 48], and in [25, 115] in relation to more general combinations than the sum of their Cheeger constants.…”

“…holding for any 𝐸 ∈ 𝒜. The Cheeger problem in this setting is standard, see [91,112], and its minimizers are now completely characterized for a large class of planar sets [41,88,92,95,114], and reasonably well-understood for convex 𝑁-dimensional bodies [5,26]. Recently, Cheeger clusters have been introduced and studied in [47], see also [30,31,48], and in [25,115] in relation to more general combinations than the sum of their Cheeger constants.…”

The Cheeger problem in abstract measure spaces

“…In particular, (P.6) holds with f (ε) = nω 1/n n ε −1/n , and follows from the standard isoperimetric inequality P Eucl (E) ≥ nω 1/n n L n (E) 1− 1 n , (7.10) holding for any E ∈ A . The Cheeger problem in this setting is standard, see [87,108] and its minimizers are now completely characterized for a large class of planar sets [38,82,88,90,111], and reasonably well understood for convex N-dimensional bodies [5,26]. Recently, Cheeger clusters have been introduced and studied in [46], see also [30,31,47].…”

Isoperimetric Sets and p-Cheeger Sets are in Bijection

“…We refer to [21] for convex sets, to [22,25] for strips, and to [26,28] for the most general statement. In dimension 2, additional properties have been proved when enjoys a rotational symmetry [7], and we also mention that a complete characterization of the Blaschke-Santaló diagram for the triplet Cheeger constant, perimeter, and area of has been recently obtained in [16], and for more general triplets in [18]. Finally, some stability results in the planar case are available in [11].…”