Containment problems in high-dimensional spaces

Abstract

For any integersn, d ≥ 2, letП(n, d) be the largest number such that every setP ofn points inR ^d contains two pointsx, y ∈ P satisfying |box_d(x, y) ∩ P| ≥П(n, d), where box_d(x, y) means the smallest closed box with sides parallel to the axes, containingx andy. We show that, for any integersn,$d \geqslant 2,\frac{2}{{(2\sqrt 2 )^{2^d } }}n + 2 \leqslant \prod (n,d) \leqslant \frac{2}{{7^{[d/5]} 2^{2^{d - 1} } }}n + 5$, which improves the lower bound due to Grolmusz [9] by a short self-contained proof.