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The $L_p$-floating area, curvature entropy, and isoperimetric inequalities on the sphere

Florian Besau, Elisabeth M. Werner

Abstract

We explore analogs of classical centro-affine invariant isoperimetric inequalities, such as the Blaschke--Santaló inequality and the $L_p$-affine isoperimetric inequalities, for convex bodies in spherical space. Specifically, we establish an isoperimetric inequality for the floating area and prove a stability result based on the spherical volume difference. The floating area has previously been studied as a natural extension of classical affine surface area to non-Euclidean convex bodies in spaces of constant curvature. In this work, we introduce the $L_p$-floating areas for spherical convex bodies, extending Lutwak's centro-affine invariant family of $L_p$-affine surface area measures from Euclidean geometry. We prove a duality formula, monotonicity properties, and isoperimetric inequalities associated with this new family of curvature measures for spherical convex bodies. Additionally, we propose a novel curvature entropy functional for spherical convex bodies, based on the $L_p$-floating area, and establish a corresponding dual isoperimetric inequality. Finally, we extend our spherical notions to space forms with non-negative constant curvature in two distinct ways. One extension asymptotically connects with centro-affine geometry on convex bodies as curvature approaches zero, while the other converges with Euclidean geometry. Notably, our newly introduced curvature entropy for spherical convex bodies emerges as a natural counterpart to both the centro-affine entropy and the Gaussian entropy of convex bodies in Euclidean space.

The $L_p$-floating area, curvature entropy, and isoperimetric inequalities on the sphere

Abstract

We explore analogs of classical centro-affine invariant isoperimetric inequalities, such as the Blaschke--Santaló inequality and the -affine isoperimetric inequalities, for convex bodies in spherical space. Specifically, we establish an isoperimetric inequality for the floating area and prove a stability result based on the spherical volume difference. The floating area has previously been studied as a natural extension of classical affine surface area to non-Euclidean convex bodies in spaces of constant curvature. In this work, we introduce the -floating areas for spherical convex bodies, extending Lutwak's centro-affine invariant family of -affine surface area measures from Euclidean geometry. We prove a duality formula, monotonicity properties, and isoperimetric inequalities associated with this new family of curvature measures for spherical convex bodies. Additionally, we propose a novel curvature entropy functional for spherical convex bodies, based on the -floating area, and establish a corresponding dual isoperimetric inequality. Finally, we extend our spherical notions to space forms with non-negative constant curvature in two distinct ways. One extension asymptotically connects with centro-affine geometry on convex bodies as curvature approaches zero, while the other converges with Euclidean geometry. Notably, our newly introduced curvature entropy for spherical convex bodies emerges as a natural counterpart to both the centro-affine entropy and the Gaussian entropy of convex bodies in Euclidean space.

Paper Structure

This paper contains 13 sections, 34 theorems, 262 equations.

Table of Contents

  1. Introduction and main results
  2. Lp-floating area and isoperimetric inequalities on the sphere
  3. Spherical curvature entropy and real-analytic extensions to space forms
  4. Special centers of spherical convex bodies
  5. Dual isoperimetric inequalities on the sphere
  6. Isoperimetric inequalities for the spherical floating area
  7. Stability for the dual volume and floating area on the sphere
  8. The Lp-floating area on the sphere
  9. Entropy of Euclidean convex bodies
  10. Entropy of spherical convex bodies
  11. Lp-floating area and Entropy and in real space forms
  12. Real-analytic extension of Lp-affine surface area to Lp-floating area and centro-affine entropy to spherical curvature entropy
  13. Real-analytic extension of Gaussian entropy to spherical curvature entropy

Key Result

Theorem 1.1

Let $d\geq 3$ and let $K\subset \mathbb{S}^d$ be a spherical convex body of class $\mathcal{C}^2_+$ such that either $K\subset C_{\mathbf{o}}(\alpha_d)$ or $K\supset C_{\mathbf{o}}(\beta_d)$, where $\tan \alpha_d=\sqrt{d(d-2)}$, $\tan\beta_d = \sqrt{d}$, and $\mathbf{o}=\mathbf{o}(K)$ is the GHS-cen with equality if and only if $K$ is geodesic ball.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Spherical Curvature Entropy
  • Theorem 1.4
  • Corollary 1.5: Positivity of spherical curvature entropy on $\mathbb{S}^2$
  • Example 2.1
  • Theorem 2.2: Gao, Hug & Schneider GHS:2003 -- GHS-center
  • proof
  • Remark 2.3: the GHS-center $\mathbf{o}(K)$ is a $H$-barycenter of the uniform measure on $K$
  • Theorem 3.1: Dual Volume Inequality (dVI) / spherical Blaschke--Santaló Inequality GHS:2003
  • ...and 65 more