Related papers: Minimum Convex Partitions and Maximum Empty Polyto…

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of…

In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the…

For convex partitions of a convex body $B$ we prove that we can put a homothetic copy of $B$ into each set of the partition so that the sum of homothety coefficients is $\ge 1$. In the plane the partition may be arbitrary, while in higher…

The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$ where $d$ is the smallest…

Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was…

We devise a polynomial-time algorithm for partitioning a simple polygon $P$ into a minimum number of star-shaped polygons. The question of whether such an algorithm exists has been open for more than four decades [Avis and Toussaint,…

In this paper, we extend and generalize several previous works on maximal-volume positions of convex bodies. First, we analyze the maximal positive-definite image of one convex body inside another, and the resulting decomposition of the…

We briefly introduce several problems: (1) a generalization of the convex fair partition conjecture, (2) on non-trivial invariants among polyhedrons that can be formed from the same set of face polygons, (3) two questions on assembling…

While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex…

Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$,…

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of at most…

We consider the following measure of symmetry of a convex n-dimensional body K: $\rho(K)$ is the smallest constant for which there is a point x in K such that for partitions of K by an n-1-dimensional hyperplane passing through x the ratio…

We consider the classical minimum and maximum cut problems: find a partition of vertices of a graph into two disjoint subsets that minimize or maximize the sum of the weights of edges with endpoints in different subsets. It is known that if…

Let $S$ be a set of $n$ points in the plane. We present several different algorithms for finding a pair of points in $S$ such that any disk that contains that pair must contain at least $cn$ points of $S$, for some constant $c>0$. The first…

This thesis consists of five papers about reduced spherical convex bodies and in particular spherical bodies of constant width on the $d$-dimensional sphere $S^d$. In paper I we present some facts describing the shape of reduced bodies of…

Approximating convex bodies is a fundamental question in geometry, which has a wide variety of applications. Given a convex body $K$ in $\textbf{R}^d$ for fixed $d$, the objective is to minimize the number of facets of an approximating…

In this paper we prove the Kneser-Poulsen conjecture for the case of large radii. Namely, if a finite number of points in Euclidean space $E^n$ is rearranged so that the distance between each pair of points does not decrease, then there…

A vast array of envy-free results have been found for the subdivision of one-dimensional resources, such as the interval $[0,1]$. The goal is to divide the space into $n$ pieces and distribute them among $n$ observers such that each…

The projective degrees of strict partitions of n were computed for all n < 101 and the partitions with maximal projective degree were found for each n. It was observed that maximizing partitions for successive values of n "lie close to each…

We present filling as a new type of spatial subdivision problem that is related to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most…