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

TL;DR

This paper investigates the minimal dimension of Euclidean space needed to embed stably $r$-neighborly submanifolds, establishing asymptotic lower bounds related to the parameters $k$ and $r$.

Contribution

It provides the first asymptotic lower bounds for the minimal embedding dimension of stably $r$-neighborly submanifolds in Euclidean spaces.

Findings

Established asymptotic lower bounds for $\Delta(k,r)$

Connected neighborliness with embedding dimension constraints

Extended understanding of geometric properties of submanifolds

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 $2kr-k$.