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

• 52A20 - Convex sets in \(n\) dimensions (including convex hypersurfaces)
• 53C42 - Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
• 57R42 - Immersions in differential topology