We generalize valuations on polyhedral cones to valuations on fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion-restriction invariants. In particular, we define a characteri
stic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion-restriction proof of a result of Klivans-Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generalization of spherical intrinsic volumes to indicator functions. We show that these intrinsic indicators yield valuations that separate polyhedral cones. Applied to hyperplane arrangements, this generalizes a result of Kabluchko on projection volumes.
We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godbersons conjecture, near-optimal bounds on M
ahler volumes, Saint-Raymond-type inequalities on mixed volumes, and reverse Kleitman inequalities for mixed volumes. We apply our results to the combinatorics of posets and prove Sidorenko-type inequalities for linear extensions of pairs of 2-dimensional posets. The results rely on some elegant decompositions of differences of anti-blocking bodies, which turn out to hold for anti-blocking bodies with respect to general polyhedral cones.
A lattice polytope is free (or empty) if its vertices are the only lattice points it contains. In the context of valuation theory, Klain (1999) proposed to study the functions $alpha_i(P;n)$ that count the number of free polytopes in $nP$ with $i$ ve
rtices. For $i=1$, this is the famous Ehrhart polynomial. For $i > 3$, the computation is likely impossible and for $i=2,3$ computationally challenging. In this paper, we develop a theory of coprime Ehrhart functions, that count lattice points with relatively prime coordinates, and use it to compute $alpha_2(P;n)$ for unimodular simplices. We show that the coprime Ehrhart function can be explicitly determined from the Ehrhart polynomial and we give some applications to combinatorial counting.
A spectral convex set is a collection of symmetric matrices whose range of eigenvalues form a symmetric convex set. Spectral convex sets generalize the Schur-Horn orbitopes studied by Sanyal-Sottile-Sturmfels (2011). We study this class of convex bod
ies, which is closed under intersections, polarity, and Minkowski sums. We describe orbits of faces and give a formula for their Steiner polynomials. We then focus on spectral polyhedra. We prove that spectral polyhedra are spectrahedra and give small representations as spectrahedral shadows. We close with observations and questions regarding hyperbolicity cones, polar convex bodies, and spectral zonotopes.
Baranys colorful generalization of Caratheodorys Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized Baranys theorem by replacing color classes with matroid constraints. In this note, we ob
tain corresponding results in tropical convexity, generalizing the tropical colorful Caratheodory Theorem of Gaubert-Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. In particular, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth.
Motivated by Grobner basis theory for finite point configurations, we define and study the class of standard complexes associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant under matroid d
uality. For the lexicographic term order, the standard complexes satisfy a deletion-contraction-type recurrence. We explicitly determine the lexicographic standard complexes for lattice path matroids using classical bijective combinatorics.
An $S$-hypersimplex for $S subseteq {0,1, dots,d}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this pape
r we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.
Interior and exterior angle vectors of polytopes capture curvature information at faces of all dimensions and can be seen as metric variants of $f$-vectors. In this context, Grams relation takes the place of the Euler-Poincare relation as the unique
linear relation among interior angles. We show the existence and uniqueness of Euler-Poincare-type relations for generalized angle vectors by building a bridge to the algebraic combinatorics of geometric lattices, generalizing work of Klivans-Swartz. We introduce flag-angles of polytopes as a geometric counterpart to flag-$f$-vectors. Flag-angles generalize the angle deficiencies of Descartes-Shephard, Grassmann angles, and spherical intrinsic volumes. Using the machinery of incidence algebras, we relate flag-angles of zonotopes to flag-$f$-vectors of graded posets. This allows us to determine the linear relations satisfied by interior/exterior flag-angle vectors.
In the chapter Magic with a Matrix in emph{Hexaflexagons and Other Mathematical
The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial topology to pr
ove a tight upper bound on the colorful simplicial depth. This implies a conjecture of Deza et al. (2006). Furthermore, we introduce colorful Gale transforms as a bridge between colorful configurations and Minkowski sums. Our colorful upper bound then yields a tight upper bound on the number of totally mixed facets of certain Minkowski sums of simplices. This resolves a conjecture of Burton (2003) in the theory of normal surfaces.
