[논문 리뷰] On Extremal Volume Projections of the Simplex and the Cube

Christos Pandis|arXiv (Cornell University)|2026. 01. 26.

Point processes and geometric inequalities인용 수 0

한 줄 요약

본 논문은 정칙 단순체의 초평면 투영 부피에 대한 닫힌 형태를 도출하고, 극값 투영 방향을 식별하며, 큐브의 극값 평면 투영을 재조명하고, Lp-투영체로 확장한다.

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.

연구 동기 및 목표

• Determine the hyperplane projection volumes of the regular simplex Delta_n and identify directions that maximize or minimize these volumes.
• Characterize extremal planar projections of the cube Q_n and establish sharp bounds including equality cases.
• Develop and explore extensions to L_p-projection bodies and related projection formulas.
• Connect projection results to width, volume, and polar/projection relationships within convex geometry.

제안 방법

• Develop a closed-form formula for the (n-1)-dimensional volume of projections: vol_{n-1}(Proj_{a^⊥∩H} Delta_n) = (1/2) * sqrt(n+1) / (n-1)! * sum_{j=1}^{n+1} |a_j| with a in the appropriate subspace.
• Identify extremal directions a by analyzing the l1-norm of a under sum-to-zero constraints (sum a_i = 0).
• Use Cauchy’s formula for polytopes to connect projection volumes to facet normals and to derive exact extremal vectors.
• Derive planar projection results for Q_n, showing vol_{n-1}(Proj_{a^⊥} Q_n) = sum_j |a_j| and establish sharp bounds with explicit equality conditions.
• Present L_p projection body generalizations: h_{Π_p Q_n}^p(a) = (1/2^{1-p}) sum_j |a_j|^p, h_{Π_p B_1^n}^p(a) = (2^{n-1}/(n-1)!) E|sum_j a_j ε_j|^p, and h_{Π_p Δ_c^n}^p(a) = ((n+1)^{(2p-1)/2})/(2(n-1)!) sum_j |a_j|^p.

실험 결과

연구 질문

• RQ1 What are the extremal (max/min) hyperplane projection volumes of the regular simplex Delta_n and which directions realize them?
• RQ2 What are the extremal planar projections (2D) of the cube Q_n and the corresponding volume bounds?
• RQ3 How do projection formulas extend to L_p projection bodies for common polytopes like Q_n, B_1^n, and centered Delta_n?
• RQ4 What is the relationship between projection volumes and width or g(DK,a) for the simplex?

주요 결과

• vol_{n-1}(Proj_{a^⊥∩H} Delta_n) = (1/2) * sqrt(n+1) /(n-1)! * sum_{j=1}^{n+1} |a_j|; extremal directions are characterized by specific half-plus/half-minus configurations.
• The minimum and maximum projection volumes are given by V_min = (1/√2) * sqrt(n+1)/(n-1)! and V_max = (1/[2(n-1)!]) * (n+1) if n is odd or sqrt{n(n+2)} if n is even, with extremal directions described.
• The cube satisfies vol_{n-1}(Proj_{a^⊥} Q_n) = sum_j |a_j|, with tight bounds 1 ≤ vol ≤ √n, attained at coordinate hyperplanes and main-diagonal orthogonals respectively. The 2D planar bound is vol_2(Proj_H Q_n) ≤ cot(π/(2n)) with equality for a regular 2n-gon projection.
• For L_p projection bodies, explicit formulas for h_{Π_p Q_n}^p(a), h_{Π_p B_1^n}^p(a), and h_{Π_p Δ_c^n}^p(a) are provided, giving a unified framework for projection volumes across p.
• The extremal width of Delta_n is achieved by the same directions that maximize the hyperplane projection volume, with g(D Delta_n,a) = (1/2) sum_j |a_j| and explicit min/max widths.