The Lower and Upper Bound Problems for Cubical Polytopes.
The -colored composition poset.
The Mayer-Vietoris and IC Equations for Convex Polytopes.
The minimal perimeter for confined deformable bubbles of equal area.
The moduli space of n tropically collinear points in Rd.
The tropical semiring (R, min, +) has enjoyed a recent renaissance, owing to its connections to mathematical biology as well as optimization and algebraic geometry. In this paper, we investigate the space of labeled n-point configurations lying on a tropical line in d-space, which is interpretable as the space of n-species phylogenetic trees. This is equivalent to the space of n x d matrices of tropical rank two, a simplicial complex. We prove that this simplicial complex is shellable for dimension...
The Newton polygon of a product of power series.
The number of vertices of a Fano polytope
Let be a Gorenstein, -factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.
The partition problem for equifacetal simplices.
The Polytope of m-Subspaces of a Finite Affine Space
The m-subspace polytope is defined as the convex hull of the characteristic vectors of all m-dimensional subspaces of a finite affine space. The particular case of the hyperplane polytope has been investigated by Maurras (1993) and Anglada and Maurras (2003), who gave a complete characterization of the facets. The general m-subspace polytope that we consider shows a much more involved structure, notably as regards facets. Nevertheless, several families of facets are established here. Then the...
The rank of extreme positive operators on polyhedral cones
The Salvetti complex and the little cubes
For a real central arrangement , Salvetti introduced a construction of a finite complex Sal which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement , the Salvetti complex Sal serves as a good combinatorial model for the homotopy type of the configuration space of points in , which is homotopy equivalent to the space of k little -cubes. Motivated by the importance of little cubes in homotopy theory, especially in the study of...
The semiregualr polytopes.
The skeleta of convex bodies
The connectivity and measure theoretic properties of the skeleta of convex bodies in Euclidean space are discussed, together with some long standing problems and recent results.
The space of tropically collinear points is shellable.
The special cuts of the 600-cell.
The toric ideal of a matroid of rank 3 is generated by quadrics.
The upper bound conjecture for arrangements of halfspaces.
The Upper Envelope of Voronoi Surfaces and Its Applications.
The Weakly Neighborly Polyhedral Maps on the 2-Manifold with Euler Characteristic -1.
Three-dimensional pseudomanifolds on eight vertices.