The Monge--Kantorovich problem, the Schur--Horn theorem, and the diffeomorphism group of the annulus

TL;DR

The paper develops a geometric bridge between the Monge–Kantorovich problem and Schur–Horn theory, first in a finite-dimensional setting via adjoint orbits of $U(n)$ and then in an infinite-dimensional diffeomorphism framework on the annulus. It shows how majorization, doubly stochastic operations, and gradient flows under the normal metric yield primal–dual inequalities $M\ge K\ge D$, and in the finite case recover the exact equalities $M=K=D$ (Brezis). Extending to ${\rm SDiff}(\mathcal{A})$ and ${\rm SMeas}(\mathcal{A})$, the work establishes a spectral rearrangement picture via the Weyl semigroup and introduces corresponding infinite-dimensional Schur–Horn results, together with dual formulations and gradient-flow interpretations. The findings provide a cohesive, geometry-driven view of optimal transport in both finite and infinite-dimensional diffeomorphism contexts, with potential implications for fluid-dynamical systems and beyond.

Abstract

First, we analyze the discrete Monge--Kantorovich problem, linking it with the minimization problem of linear functionals over adjoint orbits. Second, we consider its generalization to the setting of area preserving diffeomorphisms of the annulus. In both cases, we show how the problem can be linked to permutohedra, majorization, and to gradient flows with respect to a suitable metric.

Theorems & Definitions (25)

• Theorem 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• Theorem 2.4
• Definition 2.5
• Definition 2.6
• Theorem 2.7
• Remark 2.8
• Theorem 2.9