The multidimensional truncated Moment Problem: The Moment Cone

Let mathsfA=a1,dots,am, m∈mathbbN, be measurable functions on a measurable space (X,mathfrakA). If μ is a positive measure on (X,mathfrakA) such that int ai dμ<∞ for all i, then the sequence (int a1 dμ,dots,int am dμ) is called a moment sequence. By Richter's Theorem each moment sequence has a k-atomic representing measure with k≤ m. The set S_mathsfA 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 SA. In the case when Xsubseteq ℝn and ai∈ C1(X,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.