Stability in the Busemann-Petty and Shephard problems

TL;DR

This paper establishes linear stability results for volume comparison problems of convex bodies, specifically in the Busemann-Petty and Shephard problems, under certain geometric conditions and generalizations.

Contribution

It proves linear stability in volume comparison for convex bodies using section and projection functions, extending results to higher dimensions with new conditions.

Findings

Linear stability holds for the section function when K is an intersection body.

Linear stability holds for the projection function when L is a projection body.

Generalizations allow removing additional conditions in higher dimensions.

Abstract

A comparison problem for volumes of convex bodies asks whether inequalities $f_K(\xi)\le f_L(\xi)$ for all $\xi\in S^{n-1}$ imply that $\vol_n(K)\le \vol_n(L),$ where $K,L$ are convex bodies in $\R^n,$ and $f_K$ is a certain geometric characteristic of $K.$ By linear stability in comparison problems we mean that there exists a constant $c$ such that for every $\e>0$, the inequalities $f_K(\xi)\le f_L(\xi)+\e$ for all $\xi\in S^{n-1}$ imply that $(\vol_n(K))^{\frac{n-1}n}\le (\vol_n(L))^{\frac{n-1}n}+c\e.$ We prove such results in the settings of the Busemann-Petty and Shephard problems and their generalizations. We consider the section function $f_K(\xi)=S_K(\xi)=\vol_{n-1}(K\cap \xi^\bot)$ and the projection function $f_K(\xi)=P_K(\xi)=\vol_{n-1}(K|\xi^\bot),$ where $\xi^\perp$ is the central hyperplane perpendicular to $\xi,$ and $K|\xi^\bot$ is the orthogonal projection of $K$ to $\xi^\bot.$ In these two cases we prove linear stability under additional conditions that $K$ is an intersection body or $L$ is a projection body, respectively. Then we consider other functions $f_K,$ which allows to remove the additional conditions on the bodies in higher dimensions.