A Helly-type theorem for the dimension of the kernel of a starshaped set

Abstract:

This study will investigate the dimension of the kernel of a starshaped set, and the following result will be obtained: Let S be a compact set in some linear topological space L. For $1 \leqslant k \leqslant n$, the dimension of $\ker S$ is at least k if and only if for some $\varepsilon > 0$ and some n-dimensional flat ${F^n}$ in L, every $f(n,k)$ points of S see via S a common k-dimensional neighborhood in ${F^n}$ having radius $\varepsilon$. The number $f(n,k)$ is defined inductively as follows: \[ \begin {array}{*{20}{c}} {f(2,1) = 4,} \hfill \\ {f(n,k) = f(n - 1,k) + n + 2\quad {\text {for}}\;3 \leqslant n\;{\text {and}}\;1 \leqslant k \leqslant n - 1,} \hfill \\ {f(n,n) = n + 1.} \hfill \\ \end {array} \]