Stability results for sections of convex bodies

TL;DR

This paper establishes stability versions of known symmetry and determination results for convex bodies, showing how small deviations in conditions imply near-symmetry or near-determination, with implications for convex geometry analysis.

Contribution

It provides the first stability results for theorems on convex body symmetry and determination by fractional derivatives, extending classical results with quantitative bounds.

Findings

Stability version of Makai, Martini, and Odor's symmetry theorem.

Stability results for Koldobsky and Shane's determination theorem.

Quantitative bounds linking deviations in conditions to near-symmetry.

Abstract

It is shown by Makai, Martini, and \'Odor that a convex body $K\subset\mathbb{R}^n$, all of whose maximal sections pass through the origin, must be origin-symmetric. We prove a stability version of this result. We also discuss a theorem of Koldobsky and Shane about determination of convex bodies by fractional derivatives of the parallel section function, and establish the corresponding stability result.