Homogeneous selections from hyperplanes

Given d+1 hyperplanes h1,…,hd+1 in general position in Rd, let △(h1,…,hd+1) denote the unique bounded simplex enclosed by them. There exists a constant c(d)>0 such that for any finite families H1,…,Hd+1 of hyperplanes in Rd, there are subfamilies Hi⁎⊂Hi with |Hi⁎|⩾c(d)|Hi| and a point p∈Rd with the property that p∈△(h1,…,hd+1) for all hi∈Hi⁎.