Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space
Abstract
We establish the Alexandrov-Fenchel inequality and its modified version for arbitrary, possibly nonsmooth, horospherically convex bodies in hyperbolic space ℍn+1, n≥ 2, and prove that equality in each inequality holds if and only if the h-convex body is a geodesic ball. A key ingredient is a rigidity theorem characterizing geodesic balls via proportionality between modified support measures of adjacent orders. Its proof combines the C1,α∩ W2,k+1 approximation of the outer parallel hypersurfaces by smooth strictly h-convex hypersurfaces with the estimates for the total variation norm of the signed measure associated with the proportionality relation along the shifted curvature flows independent of initial curvature bounds. For 0≤ k≤ n−1, we construct weak flows preserving the k-th modified quermassintegral W̃k with arbitrary h-convex initial bodies as Hausdorff limits of the smooth flows. The rigidity theorem above characterizes the limit shape of the flow and gives a positive lower bound on the rate of decrease of W̃k+1 for a short time whenever the initial h-convex body is not a geodesic ball, which characterizes the equality cases in the modified Alexandrov-Fenchel inequality. For the Alexandrov-Fenchel inequality, we establish the Hausdorff stability independent of curvature bounds described by the Alexandrov-Fenchel deficits for h-convex bodies, which characterizes the equality cases in the Alexandrov-Fenchel inequality.
Related
- Curvature quotient flows and Alexandrov--Fenchel inequalities in hyperbolic space
- Newton-type inequalities for the class of modified hyperbolic p-convex functions via Riemann-Liouville fractional integrals with numerical justifications
- Alexandrov's Theorem for Integral Varifolds and Applications to Geometric Inequalities
- Alexandrov--Fenchel inequalities for convex domains in the sphere
- DISCRETE MIXED VOLUMES AND ALEXANDROV–FENCHEL-TYPE INEQUALITIES
BibTeX
@misc{zhang2026modified,
title = {{Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space}},
author = {Xinxiu Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18761},
eprint = {2610.18761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18761},
}
RIS
TY - UNPB
TI - Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space
AU - Xinxiu Zhang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.18761
UR - https://arxiv.org/abs/2610.18761
AB - We establish the Alexandrov-Fenchel inequality and its modified version for arbitrary, possibly nonsmooth, horospherically convex bodies in hyperbolic space ℍ^n+1, n≥ 2, and prove that equality in each inequality holds if and only if the h-convex body is a geodesic ball. A key ingredient is a rigidity theorem characterizing geodesic balls via proportionality between modified support measures of adjacent orders. Its proof combines the C^1,α∩ W^2,k+1 approximation of the outer parallel hypersurfaces by smooth strictly h-convex hypersurfaces with the estimates for the total variation norm of the signed measure associated with the proportionality relation along the shifted curvature flows independent of initial curvature bounds. For 0≤ k≤ n−1, we construct weak flows preserving the k-th modified quermassintegral W̃_k with arbitrary h-convex initial bodies as Hausdorff limits of the smooth flows. The rigidity theorem above characterizes the limit shape of the flow and gives a positive lower bound on the rate of decrease of W̃_k+1 for a short time whenever the initial h-convex body is not a geodesic ball, which characterizes the equality cases in the modified Alexandrov-Fenchel inequality. For the Alexandrov-Fenchel inequality, we establish the Hausdorff stability independent of curvature bounds described by the Alexandrov-Fenchel deficits for h-convex bodies, which characterizes the equality cases in the Alexandrov-Fenchel inequality.
ER -