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A solution to Bezdek's conjecture

Kostiantyn Drach, Kateryna Tatarko

TL;DR

The paper resolves Bezdek's conjecture in the λ-convex setting by showing that among all λ-convex bodies in $\mathbb{R}^n$ with fixed inradius, the λ-convex lens maximizes the mean width $V_1(K)$, with equality only for the lens; this is established via a reduction to 1-convex polytopes tangent to the inscribed ball and a facet-projection comparison harnessing a key lemma. Using λ-duality, the authors derive a dual statement for the corresponding intrinsic volumes and a partial result for the circumradius, identifying the λ-convex spindle as the unique minimizer of the mean width for fixed circumradius. Under an inball-symmetry assumption, the method extends to all intrinsic volumes, yielding $V_j(K)\le V_j(L)$ for each $j$ with equality only for the lens, thereby confirming Bezdek's conjecture in this symmetric regime. Collectively, these results advance the theory of reverse isoperimetric-type inequalities for λ-convex bodies and demonstrate a sharp lens-spindle dichotomy as a function of inradius and circumradius.

Abstract

For a given $λ>0$, a convex body in $\mathbb R^n$ is $λ$-convex if it is the intersection of (finitely or infinitely many) balls of radius $1/λ$. In this note, we show that among all $λ$-convex bodies in $\mathbb R^n$, $n \geqslant 2$, with a given inradius, the $λ$-convex lens (i.e., the intersection of two balls of radius $1/λ$) has the largest mean width. This gives an affirmative answer to the conjecture of K. Bezdek. Under an additional symmetry assumption on $λ$-convex bodies, we resolve the analogous inradius conjecture of Bezdek for arbitrary intrinsic volumes. We also establish an answer to the corresponding conjecture of K. Bezdek about the circumradius. In particular, we prove that the $λ$-convex spindle (i.e., the intersection of all balls of radius $1/λ$ containing a given pair of points) is the unique minimizer of the mean width among all $λ$-convex bodies with a fixed circumradius.

A solution to Bezdek's conjecture

TL;DR

The paper resolves Bezdek's conjecture in the λ-convex setting by showing that among all λ-convex bodies in with fixed inradius, the λ-convex lens maximizes the mean width , with equality only for the lens; this is established via a reduction to 1-convex polytopes tangent to the inscribed ball and a facet-projection comparison harnessing a key lemma. Using λ-duality, the authors derive a dual statement for the corresponding intrinsic volumes and a partial result for the circumradius, identifying the λ-convex spindle as the unique minimizer of the mean width for fixed circumradius. Under an inball-symmetry assumption, the method extends to all intrinsic volumes, yielding for each with equality only for the lens, thereby confirming Bezdek's conjecture in this symmetric regime. Collectively, these results advance the theory of reverse isoperimetric-type inequalities for λ-convex bodies and demonstrate a sharp lens-spindle dichotomy as a function of inradius and circumradius.

Abstract

For a given , a convex body in is -convex if it is the intersection of (finitely or infinitely many) balls of radius . In this note, we show that among all -convex bodies in , , with a given inradius, the -convex lens (i.e., the intersection of two balls of radius ) has the largest mean width. This gives an affirmative answer to the conjecture of K. Bezdek. Under an additional symmetry assumption on -convex bodies, we resolve the analogous inradius conjecture of Bezdek for arbitrary intrinsic volumes. We also establish an answer to the corresponding conjecture of K. Bezdek about the circumradius. In particular, we prove that the -convex spindle (i.e., the intersection of all balls of radius containing a given pair of points) is the unique minimizer of the mean width among all -convex bodies with a fixed circumradius.

Paper Structure

This paper contains 11 sections, 6 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mixed volumes.
  4. $\lambda$-convexity.
  5. Proof of Theorem \ref{['Thm:Main']}
  6. Proof of Lemma \ref{['Claim:M']}
  7. Proof of Theorem \ref{['Thm:Main2']}
  8. Proof of Theorem \ref{['Thm:MainSym']}
  9. Proof of Theorem \ref{['Thm:MainSym']}
  10. Proof of Lemma \ref{['Claim:1']}
  11. Proof of Lemma \ref{['Claim:M2']}

Key Result

Theorem A

Let $n\geqslant 2$ and let $\lambda > 0$. Let $K \subset \mathbb{R}^n$ be a $\lambda$-convex body and $L \subset \mathbb{R}^n$ be a $\lambda$-convex lens. If $r(K)=r(L)$, then with equality if and only if $K$ is a $\lambda$-convex lens.

Figures (1)

  • Figure 1: The definition of a $\lambda$-convex body

Theorems & Definitions (8)

  • Theorem A: Reverse inradius inequality for mean width
  • Theorem B: Reverse circumradius inequality for mean width
  • Theorem C: Reverse inradius inequality for intrinsic volumes under symmetry assumption
  • Remark
  • Lemma 3.1
  • Definition 5.1: Inball symmetries
  • Lemma 5.2
  • Lemma 5.3