A note on the extremal noncentral sections of the cross-polytope

TL;DR

This paper investigates extremal noncentral sections of the cross-polytope, determining minimal and maximal intersection lengths and volumes of sections by hyperplanes and slabs at fixed distances from the origin.

Contribution

It extends recent research on noncentral sections from the cube to the cross-polytope, providing new extremal length and volume bounds for such sections.

Findings

Identified minimal and maximal intersection lengths with lines at fixed distances.

Determined maximal volume of hyperplane sections at large distances.

Established minimal volume of symmetric slabs with fixed width.

Abstract

We find minimal and maximal length of intersections of lines at a fixed distance to the origin with the cross-polytope. We also find maximal volume noncentral sections of the cross-polypote by hyperplanes which are at a fixed large distance to the origin and minimal volume sections by symmetric slabs of a large fixed width. This parallels recent results about noncentral sections of the cube due to Moody, Stone, Zach and Zvavitch.