Negative moments of the support function and applications to isotropic convex bodies
Abstract
We study lower tails of support functions of isotropic convex bodies. An endpoint negative moment estimate, combined with a spherical cap argument, gives polynomial deviation estimates for the inradius of random projections in every dimension; in the unconditional class this yields the optimal order of the median inradius and shows that the cube is extremal. The same argument gives estimates for convex hulls of independent rotations and, after an additional projection, a mixed rotation-projection theorem. We also prove weighted lower and upper estimates for Minkowski averages of rotated polars and their random projections, as well as an inradius dependent estimate for the geometric distance of global averages from the Euclidean ball. A family of isotropic product cylinders shows that the latter estimate is sharp, up to absolute constants, throughout the possible range of the inradius. For the cube, an explicit negative moment computation gives a quantitative projected (m,k,n)-profile and, in full dimension, recovers the classical order n/ln(en) for bounded geometric distance. Finally, we determine the sharp volume profile of intersections of independent rotations in the unconditional isotropic class and show that the cube is extremal.
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BibTeX
@misc{hmadi2026negative,
title = {{Negative moments of the support function and applications to isotropic convex bodies}},
author = {Antonios Hmadi and Dimitris-Marios Liakopoulos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27757},
eprint = {2609.27757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27757},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Negative moments of the support function and applications to isotropic convex bodies
AU - Antonios Hmadi
AU - Dimitris-Marios Liakopoulos
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27757
UR - https://arxiv.org/abs/2609.27757
AB - We study lower tails of support functions of isotropic convex bodies. An endpoint negative moment estimate, combined with a spherical cap argument, gives polynomial deviation estimates for the inradius of random projections in every dimension; in the unconditional class this yields the optimal order of the median inradius and shows that the cube is extremal. The same argument gives estimates for convex hulls of independent rotations and, after an additional projection, a mixed rotation-projection theorem. We also prove weighted lower and upper estimates for Minkowski averages of rotated polars and their random projections, as well as an inradius dependent estimate for the geometric distance of global averages from the Euclidean ball. A family of isotropic product cylinders shows that the latter estimate is sharp, up to absolute constants, throughout the possible range of the inradius. For the cube, an explicit negative moment computation gives a quantitative projected (m,k,n)-profile and, in full dimension, recovers the classical order n/ln(en) for bounded geometric distance. Finally, we determine the sharp volume profile of intersections of independent rotations in the unconditional isotropic class and show that the cube is extremal.
ER -