A weak Krasnosel′skiĭ theorem in 𝑅^{𝑑}

Breen, Marilyn

doi:10.1090/S0002-9939-1988-0962828-3

A weak Krasnosel′skiĭ theorem in $\textbf {R}^ d$
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Proc. Amer. Math. Soc. 104 (1988), 558-562 Request permission

Abstract:

Let $S$ be a compact, locally starshaped set in ${R^d}$, and let $k$ be a fixed integer, $0 \leq k \leq d$. If every $d - k + 1$ points of $S$ are clearly visible via $S$ from a common point, then for every $k$-flat $F’$ there exists a translate $F$ of $F’$ such that the following holds: To each point $s_0$ in $S \sim F$ there correspond a point ${s_m}$ in $F$ and a polygonal path $\bigcup \left \{ {[{s_{i - 1}},{s_i}]:1 \leq i \leq m} \right \}$ in $S \cap \operatorname {aff} ({s_0} \cup F)$ with $\operatorname {dist} ({s_i},F) < \operatorname {dist} ({s_{i - 1}},F),1 \leq i \leq m$. If $k = 0$ or $k = d - 1$, then each point of $S$ sees via $S$ some point of $F$. Moreover, if $k = 1$, then $F$ can be chosen so that $F \cap S$ is convex.

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