On $r$-neighborly submanifolds in $R^N$

Abstract

A submanifold $M \subset R^N$ is $r$-neighborly if for any $r$ points in $M$ there is a hyperplane, supporting $M$ and touching it at exactly these $r$ points. We prove that the minimal dimension $\Delta(k,r)$ of the Euclidean space, containing a stably $r$-neighborly submanifold, is asymptotically not smaller than $2kr-k$.