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This thesis explores two specific topics of discrete geometry, the multitriangulations and the polytopal realizations of products, whose connection is the problem of finding polytopal realizations of a given combinatorial structure. A…
For a given convex body K in $R^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We…
In this paper we give a geometric argument for bounding the diameter of a connected compact surface (with boundary) of arbitrary codimension in Euclidean space in terms of Topping's diameter bound for closed surfaces (without boundary). The…
Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an n-point set $P$, and a collection $\mathcal{F} =…
The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and have not been studied yet in the literature. Parametric polynomial curves of low…
We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product…
The linear extension diameter of a finite poset P is the maximum distance between a pair of linear extensions of P, where the distance between two linear extensions is the number of pairs of elements of P appearing in different orders in…
A combinatorial optimization problem (COP) has a finite groundset $E(\left|E\right|=N$), a weight vector $c=\left(c^e:e\in E\right)$ and a family $T\in E$ of feasible subsets with objective to find $t\in T$ with maximal weight:…
A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely classify permutation polytopes in…
Let $P$ be a bounded polyhedron defined as the intersection of the non-negative orthant ${\Bbb R}^n_+$ and an affine subspace of codimension $m$ in ${\Bbb R}^n$. We show that a simple and computationally efficient formula approximates the…
The aim of this paper is the determination of the largest $n$-dimensional polytope with $n+3$ vertices of unit diameter. This is a special case of a more general problem proposed by Graham.
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…
Given an undirected graph $G = (V,E)$, the cut polytope $\mathrm{CUT}(G)$ is defined as the convex hull of the incidence vectors of all cuts in $G$. The 1-skeleton of $\mathrm{CUT}(G)$ is a graph whose vertex set is the vertex set of the…
Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a…
A polygon C is an intersecting polygon for a set O of objects in the plane if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the…
A weighted game or a threshold function in general admits different weighted representations even if the sum of non-negative weights is fixed to one. Here we study bounds for the diameter of the corresponding weight polytope. It turns out…
The diameter of an undirected or a directed graph is defined to be the maximum shortest path distance over all pairs of vertices in the graph. Given an undirected graph $G$, we examine the problem of assigning directions to each edge of $G$…
It is well-known that the intersection of the matching polytope with a cardinality constraint is integral [8]. We prove a similar result for the polytope corresponding to the transportation problem with market choice (TPMC) (introduced in…
The maximum gap $g(f)$ of a polynomial $f$ is the maximum of the differences (gaps) between two consecutive exponents that appear in $f$. Let $\Phi_{n}$ and $\Psi_{n}$ denote the $n$-th cyclotomic and $n$-th inverse cyclotomic polynomial,…
Let $C$ be the unit circle in $\mathbb{R}^2$. We can view $C$ as a plane graph whose vertices are all the points on $C$, and the distance between any two points on $C$ is the length of the smaller arc between them. We consider a graph…