相关论文: On f-vectors of Minkowski additions of convex poly…

We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be…

We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of…

Symmetric edge polytopes are a class of lattice polytopes constructed from finite simple graphs. In the present paper we highlight their connections to the Kuramoto synchronization model in physics -- where they are called adjacency…

In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive…

We provide an explicit Dynkin diagrammatic description of the c-vectors and the d-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface…

We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also…

Under study are some vector optimization problems over the space of Minkowski balls, i.e., symmetric convex compact subsets in Euclidean space. A typical problem requires to achieve the best result in the presence of conflicting goals;…

There exist elegant methods of aligning point clouds in $\mathbb R^3$. Unfortunately, these methods fail to generalize to the case of Minkowski space, as we will show. Instead, we propose two solutions to the following problem: given…

We consider the diagonal limit of the conformal bootstrap in arbitrary dimensions and investigate the question if physical theories are given in terms of cyclic polytopes. Recently, it has been pointed out that in $d=1$, the geometric…

This paper defines, for each convex polytope $\Delta$, a family $H_w\Delta$ of vector spaces. The definition uses a combination of linear algebra and combinatorics. When what is called exact calculation holds, the dimension $h_w\Delta$ of…

Motivated by the discrete logarithmic Minkowski problem we study for a given matrix $U\in\mathbb{R}^{n\times m}$ its cone-volume set $C_{\tt cv}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{ x\in\mathbb{R}^n :…

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the…

In this paper, we consider mixed sums of generalized polygonal numbers. Specifically, we obtain a finiteness condition for universality of such sums; this means that it suffices to check representability of a finite subset of the positive…

Equifacetal simplices, all of whose codimension one faces are congruent to one another, are studied. It is shown that the isometry group of such a simplex acts transitively on its set of vertices, and, as an application, equifacetal…

In a seminal paper "Volumen und Oberfl\"ache" (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the…

Minkowski's Theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body, and, moreover, the surface area measure determines a convex body uniquely.…

The problem of calculating exact lower bounds for the number of $k$-faces of $d$-polytopes with $n$ vertices, for each value of $k$, and characterising the minimisers, has recently been solved for $n\le2d$. We establish the corresponding…

The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent…

This article introduces the theory of Veronese polytopes, a broad generalisation of cyclic polytopes. These arise as convex hulls of points on curves with one or more connected components, obtained as the image of the rational normal curve…

We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space $L^2(\Omega)$.…