Communications in Mathematics |

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 $M^m$.

Source: arXiv.org:2301.07742

Volume: Volume 31 (2023), Issue 3 (Special issue: in memory of Sergei Duzhin)

Published on: May 9, 2023

Accepted on: April 16, 2023

Submitted on: January 21, 2023

Keywords: Mathematics - Geometric Topology,Mathematics - Differential Geometry

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