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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Sets which can be extended to $m$-convex sets
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by Marilyn Breen PDF
Proc. Amer. Math. Soc. 62 (1977), 124-128 Request permission

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 \[ {\text {conv}}\{ {P_i} \cup {P_j}\} ,\quad 1 \leqslant i < j \leqslant 3.\] 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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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 62 (1977), 124-128
  • MSC: Primary 52A20
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0430962-5
  • MathSciNet review: 0430962