A Center Transversal Theorem for an Improved Rado Depth

Abstract

A celebrated result of Dol’nikov, and of Živaljević and Vrećica, asserts that for every collection of m measures μ1, ⋯ , μm on the Euclidean space Rn+m-1 there exists a projection onto an n-dimensional vector subspace Γ with a point in it at depth at least 1 / (n+ 1) with respect to each associated n-dimensional marginal measure Γ ∗μ1, ⋯ , Γ ∗μm. In this paper we consider a natural extension of this result and ask for a minimal dimension of a Euclidean space in which one can require that for any collection of m measures there exists a vector subspace Γ with a point in it at depth slightly greater than 1 / (n+ 1) with respect to each n-dimensional marginal measure. In particular, we prove that if the required depth is 1 / (n+ 1) + 1 / (3 (n+ 1) 3) then the increase in the dimension of the ambient space is a linear function in both m and n.