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A Center Transversal Theorem for mass assignments

Omar Antolín Camarena, Jaime Calles Loperena

TL;DR

The paper extends the center transversal theorem to the setting of mass assignments on real flag manifolds, replacing marginals with mass distributions on fibers of tautological bundles. It develops the notion of mass assignments on the bundles $\,\nu_i$ over $\mathrm{Fl}(n_1, obreak \dots, obreak n_r)$ and proves a center transversal-type result with an improved Rado depth of $\frac{1}{k+1} + \frac{1}{3(k+1)^3}$, under dimension constraints that scale with the flag data. The main theorem ensures the existence of a flag and a point $x$ in the appropriate subspace $V_i$ such that all mass assignments have depth bounded below by the stated quantity, with corollaries to subspaces in Grassmannians and a generalized formulation using marginal-like constructs $\mu_i^{\Gamma}$. The approach combines the mass-assignment framework with cohomological obstructions (Stiefel–Whitney classes) on flag manifolds, extending prior center transversal results and enabling lower ambient-dimension trade-offs in linear or Grassmannian settings. This broadens partition-type results in geometric measure theory by incorporating richer bundle structures and improved depth bounds.

Abstract

In this paper, based on the ideas of Blagojević, 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 Blagojević, Karasev & Magazinov in a linear subspace of lower dimension, as long as it is contained in a high-dimensional enough ambient space.

A Center Transversal Theorem for mass assignments

TL;DR

The paper extends the center transversal theorem to the setting of mass assignments on real flag manifolds, replacing marginals with mass distributions on fibers of tautological bundles. It develops the notion of mass assignments on the bundles over and proves a center transversal-type result with an improved Rado depth of , under dimension constraints that scale with the flag data. The main theorem ensures the existence of a flag and a point in the appropriate subspace such that all mass assignments have depth bounded below by the stated quantity, with corollaries to subspaces in Grassmannians and a generalized formulation using marginal-like constructs . The approach combines the mass-assignment framework with cohomological obstructions (Stiefel–Whitney classes) on flag manifolds, extending prior center transversal results and enabling lower ambient-dimension trade-offs in linear or Grassmannian settings. This broadens partition-type results in geometric measure theory by incorporating richer bundle structures and improved depth bounds.

Abstract

In this paper, based on the ideas of Blagojević, 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 Blagojević, Karasev & Magazinov in a linear subspace of lower dimension, as long as it is contained in a high-dimensional enough ambient space.
Paper Structure (6 sections, 8 theorems, 34 equations)

This paper contains 6 sections, 8 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. An Improved Rado Depth for Mass Assignments
  3. Mass Assignments
  4. Mass Assignments Over Flag Manifolds
  5. The Center Transversal Theorem for Mass Assignments
  6. Proof of theorem \ref{['Principal thm']}

Key Result

Theorem 1.2

Let $m$, $n$, and $N$ be positive integers with $N \geq m + n -1$. For every collection of $m$ measures $\mu_1 , \ldots , \mu_m$ on $\mathbb{R}^N$, there exists an $n$-dimensional linear subspace $\Gamma$ and a point $x \in \Gamma$ such that for every $1 \leq i \leq m$

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.2: Center transversal theorem
  • Theorem 1.3: Center transversal theorem with an improved Rado depth
  • Theorem 2.1
  • Theorem 2.3
  • Corollary 2.4
  • Proposition 2.5: Center Transversal theorem for mass assignments with an improved Rado depth
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 1 more