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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Sets which can be extended to $ m$-convex sets


Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 62 (1977), 124-128
MSC: Primary 52A20
MathSciNet review: 0430962
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Abstract: Let S be a compact set in $ {R^d},{T_0} \subseteq S$. Then $ {T_0}$ lies in an m-convex subset of S if and only if every finite subset of $ {T_0}$ lies in an m-convex subset of S. For S a closed set in $ {R^d}$ and $ {T_0} \subseteq S$, let $ {T_1} = \{ P:P$ a polytope in S having vertex set in $ {T_0},\dim P \leqslant d - 1\} $. If for every three members of $ {T_1}$, at least one of the corresponding convex hulls

$\displaystyle {\text{conv}}\{ {P_i} \cup {P_j}\} ,\quad 1 \leqslant i < j \leqslant 3.$

lies in S, then $ {T_0}$ lies in a 3-convex subset of S. An analogous result holds for m-convex sets provided ker $ S \ne \emptyset $.

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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0430962-5
Keywords: m-convex sets
Article copyright: © Copyright 1977 American Mathematical Society