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 $K$.
Categories: math.MG math.GT