On Extremal Volume Projections of the Simplex and the Cube

TL;DR

This paper derives formulas for the extremal volume projections of the regular simplex and cube, providing insights into their geometric properties and generalizations within the $L_p$-projection framework.

Contribution

It introduces a closed-form formula for the extremal hyperplane volume projections of the regular simplex and revisits the extremal planar projections of the cube, extending to $L_p$-projection bodies.

Findings

Closed-form formula for simplex projections

Identification of extremal projection directions

Generalizations to $L_p$-projection bodies

Abstract

Let $\Delta_n$ and $Q_n$ denote the regular $n$-simplex of side length $\sqrt{2}$ embedded in $\mathbb{R}^{n+1}$ and the volume one cube in $\mathbb{R}^n$, respectively. We derive a closed-form formula for the hyperplane volume projections of $\Delta_n$, which also yields the directions achieving the extremal volume. Moreover, we revisit the problem of extremal planar projections of $Q_n$. In addition, we present generalizations within the framework of $L_p$-projection bodies.