A Convex Body Associated to the Busemann Random Simplex Inequality and the Petty conjecture
Julián Eduardo Haddad
TL;DR
The paper introduces a convex body N_p that encapsulates the L_p-Busemann Random Simplex Inequality (RSI) and proves an isoperimetric inequality for its polar, establishing an equivalence with RSI. It advances a functional variant of RSI, showing that a corresponding inequality for functions l_i yields extremals of Gaussian-type forms and mirrors the set-based case, via a dual mixed volume framework. A dual theory is developed by defining a dual random simplex inequality using the p-curvature and the L_p surface area Ω_p, yielding both set and functional dual inequalities and connecting p=1 to Petty’s conjecture and sharp affine Sobolev-type inequalities. The article closes with open problems centered on a dualized conjecture for tilde{I}_p and its implications for Petty-type inequalities, offering a unified dual perspective on RSI and Petty-type conjectures. The results provide simpler, non-symmetrization proofs and illuminate deep connections between random simplex inequalities, affine isoperimetric-type inequalities, and Sobolev-type functional inequalities in convex geometry.
Abstract
Given $L$ a convex body, the $L_p$-Busemann Random Simplex Inequality is closely related to the centroid body $Γ_p L$ for $p=1$ and $2$, and only in these cases it can be proved using the $L_p$-Busemann-Petty centroid inequality. We define a convex body $N_p L$ and prove an isoperimetric inequality for $(N_p L)^\circ$ that is equivalent to the $L_p$-Busemann Random Simplex Inequality. As applications, we give a simple proof of a general functional version of the Busemann Random Simplex Inequality and study a dual theory related to Petty's conjectured inequality. More precisely, we prove dual versions of the $L_p$-Busemann Random Simplex Inequality for sets and functions by means of the $p$-affine surface area measure, and we prove that the Petty conjecture is equivalent to an $L_1$-Sharp Affine Sobolev-type inequality that is stronger than (and directly implies) the Sobolev-Zhang inequality.