A note on the concurrent normal conjecture

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 [4], 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.