A weak Krasnosel′skiĭ theorem in $\textbf {R}^ d$
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- by Marilyn Breen PDF
- 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.References
- Marilyn Breen, Clear visibility and sets which are almost starshaped, Proc. Amer. Math. Soc. 91 (1984), no. 4, 607–610. MR 746099, DOI 10.1090/S0002-9939-1984-0746099-5 —, Improved Krasnosel’ skii theorems for the dimension of the kernel of a starshaped set, J. Geometry 27 (1986), 175-179.
- Marilyn Breen, Points of local nonconvexity, clear visibility, and starshaped sets in $\textbf {R}^{d}$, J. Geom. 21 (1983), no. 1, 42–52. MR 731527, DOI 10.1007/BF01918129 Eduard Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jber. Deutsch. Math. Verein. 32 (1923), 175-176.
- Alfred Horn, Some generalization of Helly’s theorem on convex sets, Bull. Amer. Math. Soc. 55 (1949), 923–929. MR 31759, DOI 10.1090/S0002-9904-1949-09309-6
- Alfred Horn and F. A. Valentine, Some properties of $L$-sets in the plane, Duke Math. J. 16 (1949), 131–140. MR 28582
- V. L. Klee Jr., On certain intersection properties of convex sets, Canad. J. Math. 3 (1951), 272–275. MR 42726, DOI 10.4153/cjm-1951-031-2 —, private communication.
- M. Krasnosselsky, Sur un critère pour qu’un domaine soit étoilé, Rec. Math. [Mat. Sbornik] N. S. 19(61) (1946), 309–310 (Russian, with French summary). MR 0020248
- Steven R. Lay, Convex sets and their applications, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 655598
- Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 558-562
- MSC: Primary 52A35; Secondary 52A30
- DOI: https://doi.org/10.1090/S0002-9939-1988-0962828-3
- MathSciNet review: 962828