Homogeneous selections from hyperplanes

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^@?.

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