Abstract
Given d + 1 hyperplanes h 1,…,h d+1 in general position in ℝd, let Δ(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 ℝd, there are subfamilies H i * ⊂H i with ¦H i *¦ ≥ c(d) ¦H i ¦ and a point p ∈ ℝd with the property that p ∈ Δ(h 1,…, h d+1) for all h i ∈ H i *.
Research of the first author was partially supported by ERC Advanced Research Grant no 267165 (DISCONV), and by Hungarian National Research Grant K 83767. The second author was partially supported by Hungarian Science Foundation EuroGIGA Grant OTKA NN 102029, by Swiss National Science Foundation Grants 200021-137574 and 200020-144531, and by NSF grant CCF-08-30272. Both authors are grateful to R. Radoičić for his valuable suggestions.
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© 2013 Scuola Normale Superiore Pisa
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Bárány, I., Pach, J. (2013). Homogeneous selections from hyperplanes. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_32
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DOI: https://doi.org/10.1007/978-88-7642-475-5_32
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