New York Journal of Mathematics
Volume 20 (2014) 1021-1041


Robert Bieri, Peter Kropholler, and Brendan Owens

On subsets of Sn whose (n+1)-point subsets are contained in open hemispheres

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Published: November 7, 2014
Keywords: Convexity, Bieri-Groves conjecture
Subject: Primary 52B99; Secondary 20J06, 20F16

We investigate the nature of subsets of spheres which satisfy a tameness condition associated with the Bieri-Groves FPm-conjecture. We find that there is a natural polyhedrality in the case of n-tame subsets of an (n-1)-sphere. In the case n=3 we establish a strong polyhedrality condition for certain maximal open 3-tame sets. Many examples are included.


The second author wishes to thank the Department of Mathematics at Cornell University, the Mittag-Leffler Institute, Djursholm, and also wishes to thank the Goethe-Universität Frankfurt for their generous support. The third author was supported in part by EPSRC grant EP/I033754/1.

Author information

Robert Bieri:
Institut für Mathematik, Goethe-Universität, Robert-Mayer-Str. 6-10 60325 Frankfurt, Germany
Department of Mathematical Sciences, Binghamton University, SUNY, Binghamton, NY 13902-6000 USA

Peter Kropholler:
Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom

Brendan Owens:
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, United Kingdom