Concurrent normals of immersed manifolds

TL;DR

This paper proves a new result about the intersection properties of normals to immersed manifolds, extending conjectures related to convex bodies and using topological conditions to guarantee multiple normals intersect at common points.

Contribution

It establishes a novel link between the topology of immersed manifolds and the intersection properties of their normals, generalizing previous conjectures and results.

Findings

Almost every normal line contains at least  + eta normals from different points.

The result applies under mild topological conditions on the manifold.

It extends classical conjectures about convex bodies to more general immersed manifolds.

Abstract

It is conjectured since long that for any convex body $K \subset \mathbb{R}^n$ 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 results of Y. Martinez-Maure, and an approach by A. Grebennikov and G. Panina, we prove the following: Let a compact smooth $m$-dimensional manifold $M^m$ be immersed in $ \mathbb{R}^n$. We assume that at least one of the homology groups $H_k(M^m,\mathbb{Z}_2)$ with $k<m$ vanishes. Then under mild conditions, almost every normal line to $M^m$ contains an intersection point of at least $\beta +4$ normals from different points of $M^m$, where $\beta$ is the sum of Betti numbers of $M^m$.