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

TL;DR

This paper proves that for dimensions greater than three, generic simple convex polytopes in Euclidean space have a point with at least 2n+4 boundary normals, addressing a longstanding problem in convex geometry.

Contribution

It establishes a piecewise-linear analogue of the $2n+4$ normals conjecture for convex bodies, extending the understanding of boundary normals in higher dimensions.

Findings

For n>3, generic simple polytopes contain a point with at least 2n+4 boundary normals.

Addresses a long-standing problem in convex geometry about normals to smooth convex bodies.

Provides a new insight into the structure of boundary normals in high-dimensional convex polytopes.

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.