Related papers: Polyhedral Gauss-Bonnet theorems and valuations

In discrete differential geometry, it is widely believed that the discrete Gaussian curvature of a polyhedral vertex star equals the algebraic area of its Gauss image. However, no complete proof has yet been described. We present an…

We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem (R. C. Heitmann, C.…

We study a combinatorial notion where given a set of lattice points one takes the set of all sums of subsets of a fixed size, and we ask if the given set comes from a convex lattice polytope whether the resulting set also comes from a…

We describe the set of possible vector valued side lengths of n-gons in thick Euclidean buildings of rank 2. This set is determined by a finite set of homogeneous linear inequalities, which we call the generalized triangle inequalities.…

We investigate some aspects of the $(2+1)$-dimensional Gauss-Bonnet black hole proposed in [1][2]. The perturbations of scalar and massless spinorial fields are studied suggesting the dynamical stability of the geometry. The field evolution…

We give a formula for the Euler characteristic of a triangulated manifold of even dimension in terms of the numbers of even-dimensional faces only. The coefficients in this formula are universal (they do not depend on the dimension of the…

In this paper, we introduce a new discretization of the Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of some dual cell of a weighted triangulation at the conic singularity. A discrete…

We present alternative postulates for Euclidean geometry whose merit is that they lead to a new class of invariants and associated geometries for real finite-dimensional unital associative algebras.

Geometry and topology are fundamental to modern condensed matter physics, but their precise connection in quantum systems remains incompletely understood. Here, we develop an analytical scheme for calculating the curvature of the quantum…

We present a general formalism for describing singular hypersurfaces in the Einstein theory of gravitation with a Gauss--Bonnet term. The junction conditions are given in a form which is valid for the most general embedding and matter…

A famous theorem of Weyl states that if $M$ is a compact submanifold of euclidean space, then the volumes of small tubes about $M$ are given by a polynomial in the radius $r$, with coefficients that are expressible as integrals of certain…

The space of polynomials in two real variables with values in a 2-dimensional irreducible module of a dihedral group is studied as a standard module for Dunkl operators. The one-parameter case is considered (omitting the two-parameter case…

Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…

I discuss the symmetry of fullerenes, viruses and geodesic domes within a unified framework of icosadeltahedral representation of these objects. The icosadeltahedral symmetry is explained in details by examination of all of these…

Given a construction of smooth homotopy class invariants of smooth immersions $M^n\to R^{n+k}$. The particular case of $k=1, n\ge 1$ is a sequence of non-zero integrals, where the $n=2$ term is the Gauss-Bonnet integral

The volume and the number of lattice points of the permutohedron P_n are given by certain multivariate polynomials that have remarkable combinatorial properties. We give several different formulas for these polynomials. We also study a more…

The aim of this note is to present two results that make the task of finding equivalent polyhedral norms on certain Banach spaces, having either a Schauder basis or an uncountable unconditional basis, easier and more transparent. The…

A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly).…

For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving mixed…

A complete classification of unimodular valuations on the set of lattice polygons with values in the spaces of polynomials and formal power series, respectively, is established. The valuations are classified in terms of their behaviour with…