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Non-central sections of the regular n-simplex

Hermann König

TL;DR

The article determines the maximal non-central hyperplane sections of the regular $n$-simplex of side $\sqrt{2}$ at a fixed distance $t$ from the centroid. It leverages a volume formula for sections, due to Dirksen, to express the $(n-1)$-dimensional section volume $A(a,t)$ in terms of the coordinates of the normal vector $a$ (with $\sum a_j=0$), and analyzes extreme points via Lagrange multipliers and monotonicity results. For $n\ge 5$, it shows that the maximal non-central sections are parallel to a face (normals along the main diagonals to vertices), i.e., attained at $a^{(1)}$; a special regime in $n=4$ allows $a^{(2)}$ to compete for smaller $t$, with a crossover at $\tilde t\approx 0.3877$. In low dimensions $n=2,3$, the extrema are fully described with explicit parameterizations of the optimizing directions and corresponding volumes. These results extend prior work on central and high-distance sections and contribute a detailed picture of non-central extremals in a classical convex-body setting.

Abstract

We show that the maximal non-central hyperplane sections of the regular n-simplex of side-length sqrt 2 at a fixed distance t to the centroid are those parallel to a face of the simplex, if $\sqrt{(n-2)/(3(n+1))} < t < \sqrt{(n-1)/(2(n+1))}$ and $n>4$. For $n=4$, the same is true in a slightly smaller range for t. This adds to a previous result for $\sqrt{(n-1)/(2(n+1))} < t < \sqrt{n/(n+1)}$. For $n=2,3$, we determine the maximal and the minimal sections for all distances t to the centroid.

Non-central sections of the regular n-simplex

TL;DR

The article determines the maximal non-central hyperplane sections of the regular -simplex of side at a fixed distance from the centroid. It leverages a volume formula for sections, due to Dirksen, to express the -dimensional section volume in terms of the coordinates of the normal vector (with ), and analyzes extreme points via Lagrange multipliers and monotonicity results. For , it shows that the maximal non-central sections are parallel to a face (normals along the main diagonals to vertices), i.e., attained at ; a special regime in allows to compete for smaller , with a crossover at . In low dimensions , the extrema are fully described with explicit parameterizations of the optimizing directions and corresponding volumes. These results extend prior work on central and high-distance sections and contribute a detailed picture of non-central extremals in a classical convex-body setting.

Abstract

We show that the maximal non-central hyperplane sections of the regular n-simplex of side-length sqrt 2 at a fixed distance t to the centroid are those parallel to a face of the simplex, if and . For , the same is true in a slightly smaller range for t. This adds to a previous result for . For , we determine the maximal and the minimal sections for all distances t to the centroid.
Paper Structure (5 sections, 10 theorems, 118 equations)

This paper contains 5 sections, 10 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. A volume formula for sections of the $n$-simplex
  3. Prerequisites
  4. Proof of Theorem 1.1
  5. Dimensions $n=2$ and $n=3$

Key Result

Theorem 1.1

Let $n \ge 5$, $\Delta^n$ be the regular $n$-simplex and assume that $t \in \mathbb{R}$ satisfies $\sqrt{\frac{n-2}{3 (n+1)}} < t \le \sqrt{\frac{n}{n+1}}$. Let $a \in S^n$ with $\sum_{j=1}^{n+1} a_j = 0$. Then $H_t(a^{(1)})$ is a maximal hyperplane section of $\Delta^n$ at distance $t$ to the cen All maximal sections at distance $t$ are parallel to a boundary $(n-1)$-face, and the orthogonal ve

Theorems & Definitions (15)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • ...and 5 more