Containment problems in high-dimensional spaces

For any integers n, d greater than or equal to 2, let II(n, d) be the largest number such that every set P of n points in R(d) contains two points x, y epsilon P satisfying \box(d)(x, y)boolean AND P\ greater than or equal to II(n, d), where box,(x, y) means the smallest closed box with sides parallel to the axes, containing x and y. We show that, for any integers n, d greater than or equal to 2, 2/(2 root 2)(2d) n+2 less than or equal to II(n,d)less than or equal to 2/7([d/5])2(2d-1) n+5, which improves the lower bound due to Grolmusz [9] by a short self-contained proof.