Clear visibility and the dimension of kernels of starshaped sets

Abstract:

This paper will use the concept of clearly visible to obtain a Krasnosel’skii-type theorem for the dimension of the kernel of a starshaped set, and the following result will be proved: For each $k$ and $n$, $1 \leqslant k \leqslant n$, let $f(n,n) = n + 1$ and $f(n,k) = 2n$ if $1 \leqslant k \leqslant n - 1$. Let $S$ be a nonempty compact set in ${R^n}$. Then for a $k$ with $1 \leqslant k \leqslant n$, dim ker $S \geqslant k$ if and only if every $f(n,k)$ points of bdry $S$ are clearly visible from a common $k$-dimensional subset of $S$. If $k = 1$ or $k = n$, the result is best possible. Moreover, if $S$ is a compact, connected, nonconvex set in ${R^2}$, then bdry $S$ may be replaced by lnc $S$ in the theorem.