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 ${\mu }_1,\dots ,{\mu }_m$ 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 $\frac{1}{n+1}$ with respect to each associated $n$-dimensional marginal measure ${\Gamma }_{\ast }{\mu }_1,\dots ,{\Gamma }_{\ast }{\mu }_m$. 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 $\frac{1}{n+1}$ with respect to each $n$-dimensional marginal measure. In particular, we prove that if the required depth is $\frac{1}{n+1}+\frac{1}{3(n+1{)}^3}$ then the increase in the dimension of the ambient space is a linear function in both $m$ and $n$.

v.2: Corrections in Sections 3 and 4 implemented, not affecting the course of the proof; v.3: Replaced with a joint paper by 3 authors with a stronger result; v.4: Final version, accepted to Discrete Comp. Geom

References in corpus (2)

• Dvoretzky type theorems for multivariate polynomials and sections of convex bodies
• Barycenters of points that are constrained to a polytope skeleton