Related papers: Maximal integral simplices with no interior intege…
We study the uniqueness of minimal liftings of cut-generating functions obtained from maximal lattice-free polyhedra. We prove a basic invariance property of unique minimal liftings for general maximal lattice-free polyhedra. This…
Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic…
Given a permutation group acting on coordinates of $\mathbb{R}^n$, we consider lattice-free polytopes that are the convex hull of an orbit of one integral vector. The vertices of such polytopes are called \emph{core points} and they play a…
We show that for every lattice packing of $n$-dimensional spheres there exists an $(n/\log_2(n))$-dimensional affine plane which does not meet any of the spheres in their interior, provided $n$ is large enough. Such an affine plane is…
We consider $d$-dimensional simplicial complexes which can be PL embedded in the $2d$-dimensional euclidean space. In short, we show that in any such complex, for any three vertices, the intersection of the link-complexes of the vertices is…
In a recent paper [TMP, 200:1 (2019), 966--984] by the authors, a series of integrable discrete autonomous equations on a square lattice with a non-standard structure of generalized symmetries is constructed. We build modified series by…
We propose a conjecture regarding the integrally closedness of lattice polytopes with large lattice lengths. We demonstrate that a lattice simplex in dimension 3 (resp. 4) with lattice length of at least 2 (resp. 3 and no edge has lattice…
Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $\mathbb{F}_q^2$ over a finite field $\mathbb{F}_q$,…
The maximal index of a Euclidean lattice L of dimension n is the maximal index of the sub-lattices of L spanned by n independent minimal vectors of L. In this paper, we prove that a perfect lattice of maximal index two not provided by a…
We introduce a pointfree theory of convergence on lattices and coframes. A convergence lattice is a lattice $L$ with a monotonic map $\lim_L$ from the lattice of filters on $L$ to $L$, meant to be an abstract version of the map sending…
In this paper we establish bounds on the number of vertices for a few classes of convex sublattice-free lattice polygons. The bounds are essential for proving the formula for the critical number of vertices of a lattice polygon that ensures…
After giving a short introduction on smooth lattice polytopes, I will present a proof for the finiteness of smooth lattice polytopes with few lattice points. The argument is then turned into an algorithm for the classification of smooth…
We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We…
Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $\mathbb{F}_q^2$ over a finite field $\mathbb{F}_q$,…
We give an upper bound for the number of points of a hypersurface over a finite field that has no lines on, in terms of the dimension, the degree, and the number of the elements of the finite field.
A theorem of Howe states that every 3-dimensional lattice polytope $P$ whose only lattice points are its vertices, is a Cayley polytope, i.e. $P$ is the convex hull of two lattice polygons with distance one. We want to generalize this…
We study the geometry of convex lattice $n$-gons with $n$ boundary lattice points and $k\geq 3$ collinear interior lattice points. We describe a process to construct a primitive lattice triangle from an edge of a convex lattice $n$-gon,…
Let X be the set of integer points in some polyhedron. We investigate the smallest number of facets of any polyhedron whose set of integer points is X. This quantity, which we call the relaxation complexity of X, corresponds to the smallest…
In this paper we solve in the positive the question of whether any finite set of integers, containing the zero, is the mapping degree set between two oriented closed connected manifolds of the same dimension. We extend this question to the…
Let A be a pre-defined set of rational numbers. We say a set of natural numbers S is an A-quotient-free set if no ratio of two elements in S belongs to A. We find the maximal asymptotic density and the maximal upper asymptotic density of…