A center transversal theorem for an improved Rado depth

TL;DR

This paper extends a classical measure partition theorem by establishing a minimal ambient space dimension that guarantees a higher depth point in a subspace for any collection of measures, with the dimension increase being linear in parameters.

Contribution

It introduces a new bound on the ambient space dimension needed for a measure depth guarantee exceeding the classical threshold.

Findings

Dimension increase is linear in m and n for the improved depth.

The paper provides a quantitative bound on the ambient space dimension.

It generalizes the center transversal theorem for measure depths.

Abstract

A celebrated result of Dol'nikov, and of \v{Z}ivaljevi\'c and Vre\'cica, asserts that for every collection of $m$ measures $\mu_1,\dots,\mu_m$ on the Euclidean space $\mathbb R^{n + m - 1}$ there exists a projection onto an $n$-dimensional vector subspace $\Gamma$ with a point in it at depth at least $\tfrac{1}{n + 1}$ with respect to each associated $n$-dimensional marginal measure $\Gamma_*\mu_1,\dots,\Gamma_*\mu_m$. 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 $\Gamma$ with a point in it at depth slightly greater than $\tfrac{1}{n + 1}$ with respect to each $n$-dimensional marginal measure. In particular, we prove that if the required depth is $\tfrac{1}{n + 1} + \tfrac{1}{3(n + 1)^3}$ then the increase in the dimension of the ambient space is a linear function in both $m$ and $n$.