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Cheeger sets for rotationally symmetric planar convex bodies

Antonio Cañete

Abstract

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

Cheeger sets for rotationally symmetric planar convex bodies

Abstract

In this note we obtain some properties of the Cheeger set asociated to a -rotationally symmetric planar convex body . More precisely, we prove that is also -rotationally symmetric and touches all the edges of .
Paper Structure (6 sections, 4 theorems, 4 equations, 5 figures)

This paper contains 6 sections, 4 theorems, 4 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some generalities on the Cheeger problem
  4. The class of $k$-rotationally symmetric planar convex bodies
  5. Main results
  6. Further comments

Key Result

Lemma 2.1

Let $\Omega$ be a planar convex body. Then, there exists a Cheeger set $C_\Omega$ of $\Omega$. Moreover,

Figures (5)

  • Figure 1: Some examples of $k$-rotationally symmetric planar convex bodies: an equilateral triangle and a Reuleaux triangle ($k=3$), a Reuleaux pentagon ($k=5$), and a circle
  • Figure 2: Two $k$-rotationally symmetric planar convex bodies, for $k=3$ and $k=5$
  • Figure 3: A 2-rotationally symmetric planar convex body
  • Figure 4: Dots and edges of two different $5$-rotationally symmetric planar convex bodies
  • Figure 5: A $3$-rotationally symmetric hexagon

Theorems & Definitions (7)

  • Lemma 2.1
  • Theorem 2.2
  • Definition 2.4
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof