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New Reverse Isoperimetric Inequalities

Scott Parkins, Glen Wheeler

Abstract

We establish three reverse inequalities for strictly convex curves and surfaces. For smooth strictly convex curves in a smooth Minkowski plane we prove an anisotropic reverse isoperimetric inequality controlled by the signed Euclidean area of the Minkowski evolute. For smooth closed strictly convex surfaces in $\mathbb R^{3}$ we prove \[ \left(\int_{M}H\,dμ\right)^{2}-16π|M| \le \frac{8π}{3}\int_{M}\frac{|A^{o}|^{2}}{\mathcal K}\,dμ, \] and relate the right-hand side to the oriented volumes of the focal maps. For smooth simple closed strictly convex curves on \(\mathbb S^{2}\) we prove \[ L^{2}-A(4π-A)\le \left(\int_γ\sqrt{1+k_g^{2}}\,ds\right)^{2}-4π^{2}, \] and in fact derive an exact nonnegative remainder formula. Equality in the spherical case holds if and only if $γ$ is a geodesic circle.

New Reverse Isoperimetric Inequalities

Abstract

We establish three reverse inequalities for strictly convex curves and surfaces. For smooth strictly convex curves in a smooth Minkowski plane we prove an anisotropic reverse isoperimetric inequality controlled by the signed Euclidean area of the Minkowski evolute. For smooth closed strictly convex surfaces in we prove and relate the right-hand side to the oriented volumes of the focal maps. For smooth simple closed strictly convex curves on we prove and in fact derive an exact nonnegative remainder formula. Equality in the spherical case holds if and only if is a geodesic circle.
Paper Structure (12 sections, 18 theorems, 156 equations)

This paper contains 12 sections, 18 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Convex Bodies and Differential Geometry of the Minkowski Plane
  3. Reverse Isoperimetric Inequality in the Minkowski Plane
  4. A Reverse Minkowski Inequality for Convex Surfaces in Euclidean Space
  5. Support-function formulas
  6. Focal maps and oriented volume
  7. A Reverse Isoperimetric Inequality for Convex Curves on the Sphere
  8. Outlook
  9. Appendix
  10. Standard results in the Minkowski plane
  11. Euclidean normal-graph formulas
  12. A spectral Poincaré inequality on $\mathbb S^{n}$

Key Result

Theorem 1

Let $M_{\mathcal{K}}^{2}$ be one of the sphere, Euclidean plane, or hyperbolic space with constant Gaussian curvature $\mathcal{K}$. Let $\gamma$ be a simple closed, piecewise-smooth curve in $M_{\mathcal{K}}^{2}$ enclosing a region $\Omega$ with area $A$, and let $L$ be the length of $\gamma$. Then Equality occurs if and only if $\gamma$ is a geodesic circle in $M_{\mathcal{K}}^{2}$.

Theorems & Definitions (38)

  • Theorem 1: Teufel1992
  • Theorem 2: groemer1996geometric, Theorem 4.3.3
  • Theorem 3: Minkowski inequality
  • Theorem 4: Anisotropic reverse isoperimetric inequality
  • Remark 5
  • Theorem 6: Reverse Minkowski Inequality
  • Theorem 7: Reverse isoperimetric inequality on $\mathbb S^{2}$
  • Proposition 8
  • proof
  • Lemma 9: Signed area of a Minkowski normal graph
  • ...and 28 more