Homogeneous selections from hyperplanes

Given d + 1 hyperplanes h(1),..., h(d+1) in general position in R-d, let Delta(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,..., Hd+1 of hyperplanes in R-d, there are subfamilies H-i* subset of H-i with vertical bar H-ivertical bar >= c(d)vertical bar H-i vertical bar and a point p is an element of R-d with the property that p is an element of Delta(h(1),..., h(d+1)) for all h(i) is an element of H-i. (C) 2013 Elsevier Inc. All rights reserved.