Homogeneous selections from hyperplanes

Authors:
Imre Bárány;János Pach
Affiliations:
Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary and Department of Mathematics, University College London, Gower Street, London WC1E 6BT, Engl ...;Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary and Chair of Combinatorial Geometry, EPFL-SB-MATHGEOM-DCG, Station 8, CH-1015 Lausanne, Switz ...
Venue:
Journal of Combinatorial Theory Series B
Year:
2014

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Abstract

Given d+1 hyperplanes h"1,...,h"d"+"1 in general position in R^d, let @?(h"1,...,h"d"+"1) denote the unique bounded simplex enclosed by them. There exists a constant c(d)0 such that for any finite families H"1,...,H"d"+"1 of hyperplanes in R^d, there are subfamilies H"i^@?@?H"i with |H"i^@?|=c(d)|H"i| and a point p@?R^d with the property that p@?@?(h"1,...,h"d"+"1) for all h"i@?H"i^@?.