2026/06/30 by Yingxiang Hu, Mohammad N. Ivaki
Mathematics · #math.AP #math.DG
arxiv created 2026/07/28 · arxiv updated 2026/07/30
We obtain weighted centro-affine Bochner formulas on spherical caps associated with smooth strictly convex hypersurfaces. As a consequence, we prove weighted Poincaré inequalities on caps and on intersections of caps for a class of weights depending on the position vector X of the hypersurface. In the unconditional case, we obtain a centro-affine Poincaré inequality with weight |X|2, which is used to prove a Brunn--Minkowski inequality for the (n+2)-nd dual quermassintegral. We also establish an L0-Brunn--Minkowski inequality for the q-th dual quermassintegral for q∈(0,n), with equality only for dilates, and an Lp-Brunn--Minkowski inequality for q=n+α whenever 0<α≤ (2p(1-p))/(2-p), which in particular covers the range q∈(n,n+6-4√(2)] for suitable p∈(0,1). These Brunn--Minkowski inequalities imply weighted centro-affine Poincaré inequalities and uniqueness results for the Lp,q-Minkowski problem in the unconditional class. Our main contribution is the introduction of a flat logarithmic centro-affine geometry on the positive orthant (0,∞)n, adapted to the multiplicative structure of the L0-sum. In this geometry, a Bochner formula yields a sharp Poincaré inequality, as well as a new proof of the centro-affine Poincaré inequality with constant n due to Kolesnikov--Milman, for unconditional bodies and unconditional functions.