A Center Transversal Theorem for mass assignments

TL;DR

This paper extends the center transversal theorem to mass assignments with improved Rado depth, using mass assignments over flag manifolds, and solves a key problem in lower-dimensional linear subspaces.

Contribution

It introduces a generalized concept of mass assignments over flag manifolds and extends the center transversal theorem accordingly, addressing a previously open problem.

Findings

Extended the center transversal theorem to mass assignments with improved Rado depth.

Solved the main problem for lower-dimensional linear subspaces within high-dimensional spaces.

Utilized mass assignments over flag manifolds to generalize existing results.

Abstract

In this paper, based on the ideas of Blagojevi\'c, Karasev & Magazinov, we consider an extension of the center transversal theorem to mass assignments with an improved Rado depth. In particular we substitute the marginal of a measure by a more general concept called a mass assignment over a flag manifold. Our results also allow us to solve the main problem proposed by Blagojevi\'c, Karasev & Magazinov in a linear subspace of lower dimension, as long as it is contained in a high-dimensional enough ambient space.