Abstract
We study the univariate moment problem of piecewise-constant density functions on the interval [0,1] and its consequences for an inference problem in population genetics. We show that, up to closure, any collection of n moments is achieved by a step function with at most n − 1 breakpoints and that this bound is tight. We use this to show that any point in the n th coalescence manifold in population genetics can be attained by a piecewise constant population history with at most n − 2 changes. Both the moment cones and the coalescence manifold are projected spectrahedra and we describe the problem of finding a nearest point on them as a semidefinite program.
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