Concurrent normals of immersed manifolds

Abstract

It is conjectured since long that for any convex body K ⊂ Rn 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 Mm be immersed in Rn. We assume that at least one of the homology groups Hk (Mm, Z2) with k < m vanishes. Then under mild conditions, almost every normal line to Mm contains an intersection point of at least β + 4 normals from different points of Mm, where β is the sum of Betti numbers of Mm.