Quantitative Quermassintegral Inequalities for Nearly Spherical Sets

TL;DR

This paper proves quantitative inequalities relating geometric measures of nearly spherical sets, providing bounds on isoperimetric deficits using asymmetry and spherical deviation, advancing understanding of geometric inequalities.

Contribution

It introduces quantitative Alexandrov-Fenchel inequalities for quermassintegrals on nearly spherical sets, linking isoperimetric deficits with asymmetry and spherical deviation.

Findings

Bound the $(k,m)$-isoperimetric deficit by the Frankael asymmetry.

Establish lower bounds on the $(k,m)$-isoperimetric deficit using spherical deviation.

Extend classical inequalities to nearly spherical geometries.

Abstract

In this paper, we establish quantitative Alexandrov-Fenchel inequalities for quermassintegrals on nearly spherical sets. In particular, we bound the $(k,m)$-isoperimetric deficit from below by the Frankael asymmetry. We also find a lower bound on the $(k,m)$-isoperimetric deficit using the spherical deviation.