TY - JOUR
T1 - Relatively convex subsets of simply connected planar sets
AU - Magazanik, Evelyn
AU - Perles, Micha A.
PY - 2007/8
Y1 - 2007/8
N2 - 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).
AB - 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).
UR - http://www.scopus.com/inward/record.url?scp=58449090049&partnerID=8YFLogxK
U2 - 10.1007/s11856-007-0058-y
DO - 10.1007/s11856-007-0058-y
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AN - SCOPUS:58449090049
SN - 0021-2172
VL - 160
SP - 143
EP - 155
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
ER -