Related papers: Packing and covering with balls on Busemann surfac…

We determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, we establish maximality by extending the Lov\'asz theta number to Cayley graphs on the special orthogonal group…

We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space $H_{\mathbb R}^n$ endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the…

We generalize the classical notion of packing a set by balls with identical radii to the case where the radii may be different. The largest number of such balls that fit inside the set without overlapping is called its {\em non-uniform…

A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has size \lfloor…

Given a point set $S=\{s_1,\ldots , s_n\}$ in the unit square $U=[0,1]^2$, an anchored square packing is a set of $n$ interior-disjoint empty squares in $U$ such that $s_i$ is a corner of the $i$th square. The reach $R(S)$ of $S$ is the set…

Suppose we put an $\epsilon$-disk around each lattice point in the plane, and then we rotate this object around the origin for a set $\Theta$ of angles. When do we cover the whole plane, except for a neighborhood of the origin? This is the…

Given a sphere of any radius $r$ in an $n$-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average…

We study the problem of covering a given point set in the plane by unit disks so that each point is covered exactly once. We prove that 17 points can always be exactly covered. On the other hand, we construct a set of 657 points where an…

As the main problem, we consider covering of a $d$-dimensional cube by $n$ balls with reasonably large $d$ (10 or more) and reasonably small $n$, like $n=100$ or $n=1000$. We do not require the full coverage but only 90\% or 95\% coverage.…

Given an integer matrix A satisfying certain regularity assumptions, we consider for a positive integer s the set F_s(A) of all integer vectors b such that the associated knapsack polytope P(A,b)={x: Ax=b, x non-negative} contains at least…

Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given $k$ there is a minimal diameter of subsets at which there exists a covering with…

Universal cover in $\mathbb{E}^{n}$ is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover…

By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if $A \subset [1,2]$ is a $(\delta,\sigma)$-set, with $|A| = \delta^{-\sigma},$ then…

Given $N$ geodesic caps on the unit sphere in $\mathbb{R}^d$, and whose total normalized surface area sums to one, what is the maximal surface area their union can cover? In this work, we provide an asymptotically sharp upper bound for an…

We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body symmetric about the origin. We also find the order of magnitude of the number of (n-1)-dimensional…

Suppose that each proper subset of a set $S$ of points in a vector space is contained in the union of planes of specified dimensions, but $S$ itself is not contained in any such union. How large can $|S|$ be? We prove a general upper bound…

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk…

We consider any Finsler metric on a closed, orientable surface of genus greater than one. H. M. Morse proved that we can associate an asymptotic direction to minimal rays in the universal cover (in the Poincar\'e disc: a point on the unit…

We consider circle packings in the plane with circles of sizes $1$, $r\simeq 0.834$ and $s\simeq 0.651$. These sizes are algebraic numbers which allow a compact packing, that is, a packing in which each hole is formed by three mutually…

If the n-dimensional unit sphere is covered by finitely many spherically convex bodies, then the sum of the inradii of these bodies is at least {\pi}. This bound is sharp, and the equality case is characterized.