Results 1 
2 of
2
HOMOGENEOUS SELECTIONS FROM HYPERPLANES
"... Abstract. 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 H ∗ i ⊂ Hi with H ∗ i  ≥ c ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. 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 H ∗ i ⊂ Hi with H ∗ i  ≥ c(d)Hi  and a point p ∈ Rd with the property that p ∈ △(h1,...,hd+1) for all hi ∈ H ∗ i. 1. The main result Throughout this paper, let H1,...,Hd+1 be finite families of hyperplanes in Rd in general position. That is, we assume that (1) no element of ∪ d+1 i=1 Hi passes through the origin, (2) any d elements have precisely one point in common, and (3) no d + 1 of them have a nonempty intersection. A transversal to these families is an ordered (d + 1)tuple h = (h1,...,hd+1) ∈ ∏d+1 i=1 Hi, where hi ∈ Hi for every i. Given hyperplanes h1,...,hd+1 ⊂ Rd in general position in Rd, there is a unique simplex denoted by △ = △(h1,...,hd+1) whose boundary is contained in ∪ d+1 1 hi. Clearly, this simplex is identical to the convex hull of the points (1.1) vi = ⋂ hj, i ∈ [d + 1] j=i where, as in the sequel, [n] stands for the set {1, 2,..., n}. Our main result is the following. Theorem 1.1. For every d ≥ 1 there is a constant c(d)> 0 with the following property. Given finite families H1,...,Hd+1 of hyperplanes in Rd in general position, there are subfamilies H ∗ i ⊂ Hi with H ∗ i  ≥ c(d)Hi  for i = 1,..., d+1 and a point p ∈ Rd such that p is contained in △(h) for every transversal h ∈ ∏d+1 i=1 Hi. It follows from the general position assumption that the simplices △(h) in Theorem 1.1 also have an interior point in common. It will be convenient to use the language of hypergraphs. Let H = H(H1,...,Hd+1) be the complete (d+1)partite hypergraph with vertex