Cyclic convex bodies and optimization moment problems

Abstract

We deal with two discrete moment problems: first, deciding when a fixed element of R d is the vector of d first moments for some discrete probability distribution on a given interval [ a, b] ( feasibility moment problem) and, second, maximizing (minimizing) a given linear combination of moments on the set of discrete probability distributions on [ a, b] whose d first moments are given ( optimization moment problem). These problems are linked with the cyclic body (which is the union of all cyclic polytopes on [ a, b]). The cyclic polytopes have been extensively studied and their combinatorial and geometric properties are noteworthy. The cyclic body also has interesting geometric properties. We totally determine its facial structure and supporting hyperplanes, and we construct an external representation by means of linear inequality systems whose coefficients are symmetric polynomials depending on parameters. These tools allow us to solve the mentioned moment problems by using linear semi-infinite programming, and we obtain a representation of non-negative polynomials over [ a, b] as well.