On Extremal Volume Projections of the Simplex and the Cube

TL;DR

The paper addresses extremal volumes of projections for the regular simplex $Δ_n$ and the unit cube $Q_n$, deriving a closed-form for hyperplane projections of $Δ_n$ and identifying the corresponding extremal directions.A key contribution is the exact formula $\operatorname{vol}_{n-1}(\operatorname{Proj}_{a^{\perp}\cap \mathcal{H}}Δ_n) = \frac{1}{2}\frac{\sqrt{n+1}}{(n-1)!}\sum_{j=1}^{n+1}|a_j|$, with extremizers given by half-plus/half-minus weight vectors, depending on parity of $n$.The work also yields sharp bounds for planar projections of $Q_n$, presents a second proof via planar sections of $B_1^n$ connected to Mahler volume, and develops generalizations to $L_p$-projection bodies with explicit formulas for $Q_n$, $B_1^n$, and centered Δ-cubes.Together, these results provide exact extremizers, connect projection theory with classical convex-geometry tools, and extend the analysis to the $L_p$ Brunn–Minkowski framework.

Abstract

Let $Δ_n$ 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 $Δ_n$, 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.

Theorems & Definitions (24)

• Conjecture 1.1: Filliman filliman1990extreme, or see nayar2023extremal
• Theorem 1.2
• Proposition 1.3
• Proposition 1.4
• Remark 1.5
• Theorem 1.6: McMullen mcmullen1984volumes; Chakerian--Filliman chakerian1986measures
• Theorem 1.7: Chakerian-Filliman
• Proposition 1.8
• Proposition 1.9
• Theorem 1.10