2024/11/03 by Florian Besau, Besau, Florian, Elisabeth M. Werner +1
Mathematics · #52A20 #53A35 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Point processes and geometric inequalities #Primary 52A55 #Secondary 28A75
paper · pdf · doi:10.48550/arxiv.2411.01631
openalex publication_date 2024/11/03 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28
We explore analogs of classical centro-affine invariant isoperimetric inequalities, such as the Blaschke--Santaló inequality and the Lp-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 Lp-floating areas for spherical convex bodies, extending Lutwak's centro-affine invariant family of Lp-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 Lp-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.