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Canal Classes and Cheeger Sets

Nico Lombardi, Christian Richter, Eugenia Saorín Gómez

Abstract

Giannopoulos, Hartzoulaki and Paouris asked in \cite{GHP} whether the best ratio between volume and surface area of convex bodies sharing a given orthogonal projection onto a fixed hyperplane is attained in the limit by a cylinder over the given projection. The answer to the question is known to be negative. In this paper, we prove a characterization of the positive answer in dimension $3$, using the Cheeger set of the common projection. A partial characterization is given in higher dimensions. We also prove that certain canal classes of convex bodies provide families of convex bodies satisfying a closely related inequality for a similar ratio.

Canal Classes and Cheeger Sets

Abstract

Giannopoulos, Hartzoulaki and Paouris asked in \cite{GHP} whether the best ratio between volume and surface area of convex bodies sharing a given orthogonal projection onto a fixed hyperplane is attained in the limit by a cylinder over the given projection. The answer to the question is known to be negative. In this paper, we prove a characterization of the positive answer in dimension , using the Cheeger set of the common projection. A partial characterization is given in higher dimensions. We also prove that certain canal classes of convex bodies provide families of convex bodies satisfying a closely related inequality for a similar ratio.

Paper Structure

This paper contains 7 sections, 24 theorems, 78 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and background
  3. On the negative answer to Question \ref{['Q1']}
  4. Aspects of a positive answer to Question \ref{['Q1']}
  5. On a (partial) positive answer to Question \ref{['Q1']} in arbitrary dimension
  6. Characterization of a positive answer for $n=3$: Theorem \ref{['th intro: n=3']}
  7. Inequality \ref{['e: 1']} under projection constraints

Key Result

Theorem A

Let $n\geq 2$, and let $u\in {\mathbb S}^{n-1}$. Let further $C\subset u^{\perp}$ be an $(n-1)$-dimensional convex body. Then the inequalities hold for every $L \in \mathcal{K}_C$. Moreover, for every $n \ge 3$, there exists an $(n-1)$-dimensional convex body $C \in \mathcal{K}^n$ such that the second inequality in ineq: bounded by Cheeger is strict.

Theorems & Definitions (43)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Proposition 2.2: Parini
  • Theorem 2.3: Sch
  • Definition 2.4: Sch
  • Theorem 2.5: Bonnesen, AAGM
  • Lemma 2.6: FGM
  • ...and 33 more