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Area and antipodal distance in convex hypersurfaces

James Dibble, Joseph Hoisington

Abstract

We establish a lower bound for the surface area of a closed, convex hypersurface in Euclidean space in terms of its displacement under continuous maps. As a result, a hypothesized lower bound for the volume of a Riemannian $n$-sphere, proved by Berger in dimension $n=2$ and disproved by Croke in dimensions $n \geq 3$, is valid for convex hypersurfaces in all dimensions. We also establish a sharp lower bound for the mean width of a convex hypersurface.

Area and antipodal distance in convex hypersurfaces

Abstract

We establish a lower bound for the surface area of a closed, convex hypersurface in Euclidean space in terms of its displacement under continuous maps. As a result, a hypothesized lower bound for the volume of a Riemannian -sphere, proved by Berger in dimension and disproved by Croke in dimensions , is valid for convex hypersurfaces in all dimensions. We also establish a sharp lower bound for the mean width of a convex hypersurface.

Paper Structure

This paper contains 6 sections, 36 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. When intrinsic distance is large relative to extrinsic distance
  3. When intrinsic displacement is comparable to extrinsic displacement
  4. Displacement, area, and mean width in convex hypersurfaces
  5. Asymptotic behavior of the constants in Theorem \ref{['main theorem']}
  6. Intrinsic versus extrinsic displacement of fixed-point-free maps

Key Result

Theorem 1.1

For each natural number $n$, there exists $h_n > 0$ such that, if $M^n$ is a closed, convex hypersurface in $\mathbb{R}^{n+1}$ with intrinsic distance $d_{M}$ and $n$-dimensional Hausdorff measure $\mathrm{Area}(M)$, $\alpha$ is a continuous map from $M$ to itself, and $\mu(\alpha) = \min_{x \in M} $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: The case when $\ell_{xy}^\perp$ is a support line for $\kappa$ in Lemma \ref{['width lower bound for curves']}.
  • Figure 2: A right circular cylinder whose area bounds the optimal result in Proposition \ref{['intrinsic-extrinsic proposition 1']} above, cf. Remark \ref{['formula for I']}.

Theorems & Definitions (72)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3: Berger1977, Proposition 2
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.2: Crofton's formula Crofton1868Santalo2004
  • Remark 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • ...and 62 more