Related papers: Covering a reduced spherical body by a disk

This article gives estimates on covering numbers and diameters of random proportional sections and projections of symmetric quasi-convex bodies in $\mathbb R$. These results were known for the convex case and played an essential role in…

Let a $R$-body be a closed set, complement of union of open balls of radius $R$ in the Euclidean space. Properties generalizing similar ones for convex sets are proved for the family of $R$-bodies; properties for the family of sets…

This paper uses Brin and Thickstun's theory of end reductions of non-compact 3-manifolds to study groups of covering translations of irreducible contractible open 3-manifolds W which are not homeomorphic to R^3. We associate to W an object…

Lattice coverings in the real plane by Minkowski balls are studied. We exploit the duality of admissible lattices of Minkowski balls and inscribed convex symmetric hexagons of these balls. An explicit moduli space of the areas of these…

This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \geq \frac{8}{\pi}R^2$. This inequality has long been conjectured for $R$ less…

A Sierpi\'nski packing in the $2$-sphere is a countable collection of disjoint, non-separating continua with diameters shrinking to zero. We show that any Sierpi\'nski packing by continua whose diameters are square-summable can be…

We prove that there exists a 1-convex surface whose universal covering does not satisfy the discrete disk property.

This paper focuses on curves and surfaces of constant width, with some additional results about general ovals. We emphasize the use of Fourier series to derive properties, some of which are known. Amongst other results, we show that the…

The goal of this paper is to present a lower bound for the Mahler volume of at least 4-dimensional symmetric convex bodies. We define a computable dimension dependent constant through a 2-dimensional variational (max-min) procedure and…

By taking into account the local energy balance per unit volume between the viscous heating and the advective cooling plus the radiative cooling, we investigate the vertical structure of radiation pressure-supported accretion disks in…

This paper is a companion paper to [Lipman and Daubechies 2011]. We provide numerical procedures and algorithms for computing the alignment of and distance between two disk type surfaces. We provide a convergence analysis of the discrete…

In this note, we give a new and simple proof of a result in {\cite{DX1}} which states that any smooth complete self-shrinker in $\mathbb{R}^3$ with second fundamental form of constant length must be a generalized cylinder $\mathbb{S}^k…

We prove that for every centrally symmetric convex polygon Q, there exists a constant alpha such that any alpha*k-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound…

Let $D$ be a Riemannian 2-disc of area $A$, diameter $d$ and length of the boundary $L$. We prove that it is possible to contract the boundary of $D$ through curves of length $\leq L + 200d\max\{1,\ln {\sqrt{A}\over d} \}$. This answers a…

We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in John's position, as well as for their polar bodies. These estimates extend some well-known results for convex…

Consider the problem of fnding the smallest area convex $k$-gon containing $n\in\mathbb{N}$ congruent disks without an overlap. By using Wegner inequality in sphere packing theory we give a lower bound for the area of such polygons. For…

For two planar convex bodies, $C$ and $D$, consider a packing $S$ of $n$ positive homothets of $C$ contained in $D$. We estimate the total perimeter of the bodies in $S$, denoted ${\rm per}(S)$, in terms of ${\rm per}(D)$ and $n$. When all…

A system of sets forms an {\em $m$-fold covering} of a set $X$ if every point of $X$ belongs to at least $m$ of its members. A $1$-fold covering is called a {\em covering}. The problem of splitting multiple coverings into several coverings…

We prove that every distinct covering system has a modulus divisible by either 2 or 3.

A circle C holds a convex body K if C does not meet the interior of K and if there does not exist any euclidean displacement which moves C as far as desired from K, avoiding the interior of K. The purpose of this note is to explore how…