Relatively convex subsets of simply connected planar sets

Evelyn Magazanik*, Micha A. Perles

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Let D ⊂ R2 be simply connected. A subset K ⊂ D is relatively convex if a, b K, [a, b] K ⊂ D implies [a, b] K ⊂ K. We establish the following version of Helly's Topological Theorem: If K is a family of (at least 3) compact, polygonally connected and relatively convex subsets of D, then ∩ K ≠ 0, provided each three members of K meet. We also prove other results related to the combinatorial metric ρK(a, b) (= smallest number of edges of a polygonal path from a to b in K).

Original languageEnglish
Pages (from-to)143-155
Number of pages13
JournalIsrael Journal of Mathematics
Volume160
DOIs
StatePublished - Aug 2007

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