Sections of convex bodies in John's and minimal surface area position

TL;DR

This paper extends known geometric estimates to lower-dimensional sections of convex bodies in John's and minimal surface area positions, providing new bounds for volume, mean width, and related functionals.

Contribution

It introduces new estimates for sections of convex bodies in John's and minimal surface area positions, expanding results beyond the cube and simplex cases.

Findings

New bounds for volume and mean width of sections in John's position

Estimates for the Wills functional of sections and polar bodies

Results for centrally symmetric convex bodies in minimal surface area position

Abstract

We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in John's position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in John's position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.