On the concurrent normals conjecture for convex bodies

Abstract

It is conjectured that any convex body in R n $\mathbb {R}^{n}$ has an interior point lying on normals through 2n distinct boundary points. This concurrent normals conjecture has been proved for n = 2 $n=2$ and n = 3 $n= 3$ by Heil. Pardon put forward a proof for n = 4 $n=4$ . For n ⩾ 5 $n\geqslant 5$ , it is only known that any convex body in R n $\mathbb {R}^{n}$ has an interior point lying on normals through six distinct boundary points. For n ∈ { 3 , 4 } $n\in \lbrace 3,4\rbrace $ , we prove that any normal through a boundary point to any convex body K (with a smooth enough support function) in R n $\mathbb {R}^{n}$ passes arbitrarily close to the set of interior points of K ∪ L $K\cup L$ lying on normals through at least six distinct points of ∂ K $\partial K$ , where L is the body bounded by the smallest convex parallel hypersurface to ∂ K $\partial K$ whose unit normal points in the opposite direction. This study leads us to introduce and study new concepts for studying focals of closed convex hypersurfaces in R n + 1 $\mathbb {R} ^{n+1}$ . Finally, we prove that for some convex body K of R 4 $\mathbb {R}^{4}$ , there are only six normal lines passing through the center of the minimal spherical shell.