A note on the concurrent normal conjecture

TL;DR

This paper discusses a long-standing conjecture about the existence of a point inside convex bodies that lies on many normals from the boundary, providing a new proof for a related result in higher dimensions.

Contribution

It offers a short proof that, under mild conditions, almost every normal in higher-dimensional convex bodies intersects at least six other normals, advancing understanding of the normal conjecture.

Findings

Almost every normal contains at least six normals from different boundary points.

The result holds for convex bodies in dimensions three and higher.

The proof is inspired by recent work of Y. Martinez-Maure.

Abstract

It is conjectured since long that for any convex body $K \in \mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at least $2n$ normals from different points on the boundary of $K$. The conjecture is known to be true for $n=2,3,4$. Motivated by a recent preprint of Y. Martinez-Maure, we give a short proof of his result: for dimension $n\geq 3$, under mild conditions, almost every normal through a boundary point to a smooth convex body $K\in \mathbb{R}^n$ contains an intersection point of at least $6$ normals from different points on the boundary of $K$.