The multidimensional truncated Moment Problem: The Moment Cone

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.