On the concurrent normals conjecture for convex bodies

Yves Martinez-Maure (1, 2)

1 SU - Sorbonne Université
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

It is conjectured that any convex body in R^n has an interior point lying on normals through 2n distinct boundary points. This concurrent normals conjecture has been proved for n = 2 and n = 3 by E. Heil. J. Pardon put forward a proof for n = 4. For n>4, it is only known that any convex body in R^2n has an interior point lying on normals through six distinct boundary points. For n=3 ou 4, we prove in this paper that any normal through a boundary point to any convex body K (with a smooth enough support function) in R^n passes arbitrarily close to the set of interior points of K lying on normals through at least 6 distinct points of the boundary. This study leads us to introduce and study new concepts for studying focals of closed convex hypersurfaces in R^(n+1). Finally, we prove that for some convex body K of R^4 , there are only 6 normal lines passing through the center of the minimal spherical shell.

Domaines: Mathématiques [math]

Yves Martinez-Maure. On the concurrent normals conjecture for convex bodies. 2022. ⟨hal-03292275v5⟩