Related papers: The harmonic polytope
A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete…
We provide formulas and algorithms for computing the excess numbers of certain ideals. The solution for monomial ideals is given by the mixed volumes of certain polytopes. These results enable us to design specific homotopies for numerical…
The cosmohedron was recently proposed as a polytope underlying the cosmological wavefunction for $\text{Tr}(\Phi^3)$ theory. Its faces were conjectured to be in bijection with Matryoshkas, which are obtained from a subdivision of a polygon…
Recent breakthroughs in the study of scattering amplitudes have uncovered profound and unexpected connections with combinatorial geometry. These connections range from classical structures -- such as polytopes, matroids, and Grassmannians…
We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of…
An orbit polytope is the convex hull of an orbit under a finite group $G \leq \operatorname{GL}(d,\mathbb{R})$. We develop a general theory of possible affine symmetry groups of orbit polytopes. For every group, we define an open and dense…
Flow polytopes of acyclic oriented graphs arise naturally in combinatorial optimization, and the study of their volumes and triangulations has revealed intriguing connections across combinatorics, geometry, algebra, and representation…
A generic orthotope is an orthogonal polytope whose tangent cones are described by read-once Boolean functions. The purpose of this note is to develop a theory ofEhrhart polynomials for integral generic orthotopes. The most remarkable part…
The Minkowski mixed volume of $n$ subpolytopes $D_1, \dots, D_n$ of a polytope $P \subset {\mathbb R}^n$ clearly does not exceed the normalized volume $n! \text{Vol}(P)$. Equality holds if and only if the subpolytopes are interlaced, i.e.,…
The study of the combinatorial diameter of a polyhedron is a classical topic in linear-programming theory due to its close connection with the possibility of a polynomial simplex-method pivot rule. The 2-sum operation is a classical…
Hladky, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs. Combinatorial optimization studies the…
The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another…
Recently, a combinatorial interpretation of Baldoni and Vergne's generalized Lidskii formula for the volume of a flow polytope was developed by Benedetti et al.. This converts the problem of computing Kostant partition functions into a…
We investigate the convex hulls of the eight dihedral symmetry classes of $n \times n$ alternating sign matrices, i.e., ASMs invariant under a subgroup of the symmetry group of the square. Extending the prefix-sum description of the ASM…
An infinite family of Boolean polynomials which correspond to the discrete average maps, defined in [2], is constructed and their algebraic and combinatorial properties are investigated. They turn out to be balanced, and some recurrence…
The geometry of the dual amplituhedron is generally described in reference to a particular triangulation. A given triangulation manifests only certain aspects of the underlying space while obscuring others, therefore understanding this…
A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by…
We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a…
Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial…
We study the topology of the space of harmonic maps from $S^2$ to \CP 2$. We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for…