On the volume of projections of the cross-polytope

TL;DR

This paper investigates the volume of projections of the n-dimensional cross-polytope, establishing maximum volumes for certain dimensions and identifying local maxima that are not global, contributing to geometric projection theory.

Contribution

It proves maximum projection volumes for 2D and 3D cases and identifies local maxima in the volume of projections for any n > k ≥ 2.

Findings

Maximum volume projections onto 2D and 3D subspaces

Exact lower bounds for 2D projections

Existence of local maxima that are not global

Abstract

We study properties of the volume of projections of the $n$-dimensional cross-polytope $\crosp^n = \{ x \in \R^n \mid |x_1| + \dots + |x_n| \leqslant 1\}.$ We prove that the projection of $\crosp^n$ onto a $k$-dimensional coordinate subspace has the maximum possible volume for $k=2$ and for $k=3.$ We obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global ones for the volume of a projection of $\crosp^n$ onto a $k$-dimensional subspace for any $n > k \geqslant 2.$