The 𝐿 𝑝 -floating area, curvature entropy, and isoperimetric inequalities on the sphere

We explore analogs of classical centro-affine invariant isoperimetric inequalities, such as the Blaschke–Santaló inequality and the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> Lₚ -floating areas for spherical convex bodies, extending Lutwak’s centro-affine invariant family of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> 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.