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

TL;DR

This paper extends classical isoperimetric inequalities to spherical convex bodies by introducing $L_p$-floating areas, establishing duality, monotonicity, and stability results, and proposing a new curvature entropy functional with extensions to non-negative curvature spaces.

Contribution

It introduces the $L_p$-floating area for spherical convex bodies, extending Euclidean $L_p$-affine surface measures, and develops associated inequalities and a novel curvature entropy functional.

Findings

Established an isoperimetric inequality for spherical floating area.

Proved a duality formula and monotonicity properties for $L_p$-floating areas.

Extended notions to space forms with non-negative curvature.

Abstract

We explore analogs of classical centro-affine invariant isoperimetric inequalities, such as the Blaschke--Santal\'o 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.