An Improvement on the Rado Bound for the Centerline Depth

Abstract

Let $$\mu $$ be a Borel probability measure in $$\mathbb {R}^d$$ . For a k-flat $$\alpha $$ consider the value $$\inf \mu (H)$$ , where H runs through all half-spaces containing $$\alpha $$ . This infimum is called the half-space depth of $$\alpha $$ . Bukh, Matousek and Nivasch conjectured that for every $$\mu $$ and every $$0 \le k < d$$ there exists a k-flat with the depth at least $$\tfrac{k + 1}{k + d + 1}$$ . The Rado Centerpoint Theorem implies a lower bound of $$\tfrac{1}{d + 1 - k}$$ (the Rado bound), which is, in general, much weaker. Whenever the Rado bound coincides with the bound conjectured by Bukh, Matousek and Nivasch, i.e., for $$k = 0$$ and $$k = d - 1$$ , it is known to be optimal. In this paper we show that for all other pairs (d, k) one can improve on the Rado bound. If $$k = 1$$ and $$d \ge 3$$ we show that there is a 1-dimensional line with the depth at least $$\tfrac{1}{d} + \tfrac{1}{3d^3}$$ . As a corollary, for all (d, k) satisfying $$0< k < d - 1$$ there exists a k-flat with depth at least $$\tfrac{1}{d + 1 - k} + \tfrac{1}{3(d + 1 - k)^3}$$ .