Support Sets in Exponential Families and Oriented Matroid Theory

The closure of a discrete exponential family is described by a finite set of equations corresponding to the circuits of an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they need not be polynomial in the general case. This description allows for a combinatorial study of the possible support sets in the closure of an exponential family. If two exponential families induce the same oriented matroid, then their closures have the same support sets. Furthermore, the positive cocircuits give a parameterization of the closure of the exponential family.Comment: 27 pages, extended version published in IJA

Pattern Recognition on Oriented Matroids: Halfspaces, Convex Sets and Tope Committees

The principle of inclusion-exclusion is applied to subsets of maximal covectors contained in halfspaces of a simple oriented matroid and to convex subsets of its ground set for enumerating tope committees.Comment: 15 pages; v.2 - minor improvements, v.3,4 - new Section 7 and references adde

The universality theorem for neighborly polytopes

In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of an even-dimensional neighborly polytope. This in particular provides the final step for Mn\"ev's proof of the universality theorem for simplicial polytopes.Comment: 5 pages, 1 figure. Small change

Inequalities for the h- and flag h-vectors of geometric lattices

We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if D(L) is the order complex of a rank (r+1) geometric lattice L, then for all i \leq r/2 the h-vector of D(L) satisfies h(i-1) \leq h(i) and h(i) \leq h(r-i). We also obtain several inequalities for the flag h-vector of D(L) by analyzing the weak Bruhat order of the symmetric group. As an application, we obtain a zonotopal cd-analogue of the Dowling-Wilson characterization of geometric lattices which minimize Whitney numbers of the second kind. In addition, we are able to give a combinatorial flag h-vector proof of h(i-1) \leq h(i) when i \leq (2/7)(r + 5/2).Comment: 15 pages, 2 figures. Typos fixed; most notably in Table 1. A note was added regarding a solution to problem 4.

Universality theorems for inscribed polytopes and Delaunay triangulations

We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to M\"obius transformation) of a polytope. If the semialgebraic set is moreover open, then, in addition, we prove that (up to homotopy) it is a retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of $\mathbb{Q}$ are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mn\"ev universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, our results imply that the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.Comment: 15 pages, 2 figure