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On $2n+4$ normals conjecture for convex polytopes in $\mathbb{R}^n$

Ivan Nasonov, Gaiane Panina

TL;DR

The paper proves that for every generic simple polytope $\mathbf{P}\subset \mathbb{R}^n$ with $n>2$, there exists an interior point from which at least $2n+4$ normals emanate to the boundary. The method reframes normal directions as critical points of the squared-distance function $SQD^{\mathbf{P}}_y$ and leverages a bifurcation-set framework together with spherical-geometry tools to constrain face configurations. The $n=4$ case is treated with a dedicated spherical-link analysis, while for $n>4$ a combinatorial obstruction on face adjacencies yields a contradiction, proving the bound in all higher dimensions. This work provides a piecewise-linear analogue of the classical smooth normals conjecture and advances understanding of normal structure in polytopes via Morse-theoretic and geometric arguments.

Abstract

We prove that for $n>3$ each generic simple polytope in $\mathbb{R}^n$ contains a point with at least $2n+4$ emanating normals to the boundary. This result is a piecewise-linear counterpart of a long-standing problem about normals to smooth convex bodies.

On $2n+4$ normals conjecture for convex polytopes in $\mathbb{R}^n$

TL;DR

The paper proves that for every generic simple polytope with , there exists an interior point from which at least normals emanate to the boundary. The method reframes normal directions as critical points of the squared-distance function and leverages a bifurcation-set framework together with spherical-geometry tools to constrain face configurations. The case is treated with a dedicated spherical-link analysis, while for a combinatorial obstruction on face adjacencies yields a contradiction, proving the bound in all higher dimensions. This work provides a piecewise-linear analogue of the classical smooth normals conjecture and advances understanding of normal structure in polytopes via Morse-theoretic and geometric arguments.

Abstract

We prove that for each generic simple polytope in contains a point with at least emanating normals to the boundary. This result is a piecewise-linear counterpart of a long-standing problem about normals to smooth convex bodies.
Paper Structure (6 sections, 8 theorems, 2 equations)

This paper contains 6 sections, 8 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Some necessary tools from spherical geometry
  4. Skew and nice faces
  5. Proof of Theorem \ref{['Thmmain']} for $n=4$
  6. Proof of Theorem \ref{['Thmmain']} for $n>4$

Key Result

Theorem 1

For $n>2$, each generic simple polytope $\mathbf{P} \in \mathbb{R}^n$ has a point with (at least) $2n+4$ emanating normals to the boundary.

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 1
  • Definition 1
  • Lemma 2
  • Definition 2
  • Definition 3
  • Lemma 3
  • proof
  • Proposition 1
  • proof
  • ...and 6 more