Related papers: Normally located polyhedra
In 1970, Coxeter gave a short and elegant geometric proof showing that if $p_1, p_2, \ldots, p_n$ are vertices of an $n$-gon $P$ in cyclic order, then $P$ is affinely regular if, and only if there is some $\lambda \geq 0$ such that…
We consider two sequences of orthogonal polynomials $(P_n)_{n\geq 0}$ and $(Q_n)_{n\geq 0}$ such that $$ \sum_{j=1} ^{M} a_{j,n}\mathrm{S}_x\mathrm{D}_x ^k P_{k+n-j} (z)=\sum_{j=1} ^{N} b_{j,n}\mathrm{D}_x ^{m} Q_{m+n-j} (z)\;, $$ with…
We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well…
We adapt an argument of Tao and Vu to show that if $\lambda_1\le\cdots\le\lambda_d$ are the successive minima of an origin-symmetric convex body $K$ with respect to some lattice $\Lambda<\mathbb{R}^d$, and if we set…
The first-order correction of the perturbative solution of the coupled equations of the quadratic gravity and nonlinear electrodynamics is constructed, with the zeroth-order solution coinciding with the ones given by Ay\'on-Beato and…
We study the Minkowski length L(P) of a lattice polytope P, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P. The Minkowski length represents the largest possible number of factors in…
In this paper, we show that Oda's question holds for $n$-dimensional simplicial reflexive polytope $P$ and lattice polytope $Q$ containing the origin, when the vertex of $Q$ is either a vertex of $P$ or the origin, provided that $P$ has no…
We consider the problem of finding the minimum of inhomogeneous Gaussian lattice sums: Given a lattice $L \subseteq \mathbb{R}^n$ and a positive constant $\alpha$, the goal is to find the minimizers of $\sum_{x \in L} e^{-\alpha \|x -…
The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with…
A convex polyhedron, that is, a compact convex subset of $\mathbb{R}^3$ which is the intersection of finitely many closed half-spaces, can be rectified by taking the convex hull of the midpoints of the edges of the polyhedron. We derive…
Given a proper cone $K \subseteq \mathbb{R}^n$, a multivariate polynomial $f \in \mathbb{C}[z] = \mathbb{C}[z_1, \ldots, z_n]$ is called $K$-stable if it does not have a root whose vector of the imaginary parts is contained in the interior…
We prove that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons. This solves the discrete version of the Heine-Shephard problem for two bodies in the plane. As an…
In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed ${\rm PGL}_2$-local systems on a punctured surface $S$. The moduli space is birational to a cluster $\mathcal{X}$-variety,…
For each pair of coprime integers $a$ and $b$ we have a rational $q$-Catalan number $\operatorname{Cat}(a,b)_q=\binom{a+b}{a}_q/[a+b]_q$. It is known that this is a polynomial in $q$ with nonnegative integer coefficients, but the nature of…
Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the…
Given a pure, full-dimensional, locally strongly connected polyhedral complex C with convex support, we characterize, by a local codimension-2 condition, polyhedral complexes that coarsen C. The proof of the characterization draws upon a…
Let $\Delta$ be an $n$-dimensional lattice polytope. The smallest non-negative integer $i$ such that $k \Delta$ contains no interior lattice points for $1 \leq k \leq n - i$ we call the degree of $\Delta$. We consider lattice polytopes of…
The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a…
We show that the following classes of lattice polytopes have unimodular covers, in dimension three: the class of parallelepipeds, the class of centrally symmetric polytopes, and the class of Cayley sums $\text{Cay}(P,Q)$ where the normal…
In this paper, we study the properties of the Fermat-Weber point for a set of fixed points, whose arrangement coincides with the vertices of a regular polygonal chain. A $k$-chain of a regular $n$-gon is the segment of the boundary of the…