Related papers: Finitely generated dyadic convex sets

In this article we prove some previously announced results about metric ultraproducts of finite simple groups. We show that any non-discrete metric ultraproduct of alternating or special linear groups is a geodesic metric space. For more…

The non-empty finite subsets of a multiplicatively written monoid form a monoid under setwise multiplication. The same holds for finite subsets containing the identity element. Partly due to their unusual arithmetic properties, these…

A perfect cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. The existence of such cuboids is neither proved, nor disproved. A rational perfect cuboid is a natural…

A convex set with nonempty interior is maximal lattice-free if it is inclusion-maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision…

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric…

In this paper we define and investigate a class of polytopes which we call "vertex generated" consisting of polytopes which are the average of their $0$ and $n$ dimensional faces. We show many results regarding this class, among them: that…

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any $d\ge 3$, the graph of a cubical $d$-polytope…

We consider groups defined by non-empty balanced presentations with the property that each relator is of the form R(x,y), where x and y are distinct generators and R(.,.) is determined by some fixed cyclically reduced word R(a,b) that…

In this paper we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two if and only if each open face…

A polytrope is a tropical polytope which at the same time is convex in the ordinary sense. A $d$-dimensional polytrope turns out to be a tropical simplex, that is, it is the tropical convex hull of $d+1$ points. This statement is equivalent…

In the spirit of the Genetics of the Regular Figures, by L. Fejes T\'oth, we prove the following theorem: If $2n$ points are selected in the $n$-dimensional Euclidean ball $B^n$ so that the smallest distance between any two of them is as…

The class of convex sets that admit approximations as Minkowski sum of a compact convex set and a closed convex cone in the Hausdorff distance is introduced. These sets are called approximately Motzkin-decomposable and generalize the notion…

We study the Poincar\'e series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we…

A dimension group is a partially ordered countable group such that (1) every finite subset is contained in an ordered subgroup which is a finite direct power of Z and (2) the group has an order unit i.e. a positive element u such that every…

Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a…

A space is defined to be "$n$-spheroidal" if it has the homotopy type of an $n$-dimensional CW-complex $X$ with $H_{n}(X, \mathbb{Z})$ not zero and finitely generated. A group $G$ is called "$n$-spheroidal" if its classifying space $K(G,1)$…

Random simplices and more general random convex bodies of dimension $p$ in $\mathbb{R}^n$ with $p\leq n$ are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if…

We study the question whether the affine semigroup of integer points in a convex cone can be finitely generated up to symmetries of the cone. We establish general properties of finite generation up to symmetry, and then concentrate on the…

In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper studies semiregular abstract polytopes, which have abstract regular facets, still…

A lattice Delaunay polytope is known as perfect if the only ellipsoid, that can be circumscribed about it, is its Delaunay sphere. Perfect Delaunay polytopes are in one-to-one correspondence with arithmetic equivalence classes of positive…