The multidimensional truncated Moment Problem: The Moment Cone

TL;DR

This paper investigates the structure of the multidimensional truncated moment cone, focusing on its facial structure, bounds on the Carathéodory number, and differential properties of the moment map, with applications to related problems.

Contribution

It provides a detailed analysis of the facial structure, Carathéodory bounds, and differential properties of the multidimensional moment cone, extending understanding of its geometric and regularity features.

Findings

Characterization of exposed faces and facial dimensions of the moment cone.

Bounds on the Carathéodory number for representing measures.

Analysis of the differential structure and regularity of the moment map.

Abstract

Let $\mathsf{A}=\{a_1,\dots,a_m\}$, $m\in\mathbb{N}$, be measurable functions on a measurable space $(\mathcal{X},\mathfrak{A})$. If $\mu$ is a positive measure on $(\mathcal{X},\mathfrak{A})$ such that $\int a_i d\mu<\infty$ for all $i$, then the sequence $(\int a_1 d\mu,\dots,\int a_m d\mu)$ is called a moment sequence. By Richter's Theorem each moment sequence has a $k$-atomic representing measure with $k\leq m$. The set $\mathcal{S}_\mathsf{A}$ of all moment sequences is the moment cone. The aim of this paper is to analyze the various structures of the moment cone. The main results concern the facial structure (exposed faces, facial dimensions) and lower and upper bounds of the Carath\'eodory number (that is, the smallest number of atoms which suffices for all moment sequences) of the convex cone $\mathcal{S}_{\mathcal{A}}$. In the case when $\mathcal{X}\subseteq \mathbb{R}^n$ and $a_i\in C^1(\mathcal{X},\mathbb{R})$, the differential structure of the moment map and regularity/singularity properties of moment sequences are analyzed. The maximal mass problem is considered and some applications to other problems are sketched.