# Questions tagged [packing-and-covering]

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### On the thinnest cover of the plane by a given planar convex region

Is the following claim valid? Claim: Given any planar convex region C, the thinnest cover of the plane with copies of C cannot have any region where more than 2 copies overlap. In general, the ...
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### Lower bound estimate for the sum $\sum \text{diam}(U)^d$ over all countable covers of a cube

This question is inspired from the definition of Hausdorff measure. Let $C$ be a closed unit hypercube in $\mathbb R^d$ (side length equal to one, including boundary. The cube itself is at top ...
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### Can Chang and Wang's proof of Thue’s Theorem on circular packing be extended into other dimentions?

The simplicity of Chang and Wang's proof of Thue’s Theorem (link on arxiv) on circular packing took me by surprise. Have similar ideas been found helpful in other dimensions? For example, partition ...
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Let $k$, $m$ and $n$ be positive integers. Let $r$ be a positive real number. The $n$-th ordered $r$-disk configuration space on the Euclidean space $\mathbb{R}^{mk}$ is $$F_r(\mathbb{R}^{mk},... 0 votes 1 answer 115 views ### packing numbers and configuration spaces of the torus Let S^1 be the unit circle of radius 1. For any k\geq 1, let the k-dimensional torus T^k= \underbrace{S^1\times S^1\times\cdots\times S^1}_k be the k-fold self-Cartesian ... 2 votes 0 answers 132 views ### Covering number bound for the Sobolev space W^{1,1}(E), with E \subset \mathbb{R}^n Let E \subset \mathbb{R}^n have finite diameter D(E)= \sup_{x,x' \in E}\Vert x-x' \Vert_n. Let W^{1,1}(E) denote the class of functions f on E admitting integrable first order weak ... 1 vote 1 answer 248 views ### Covering number in the space of symmetric matrices Let S_n(\mathbb{R}) be the set of symmetric matrices of size n \times n. Note \|\Theta\|_{0} the number of nonzero elements of a matrix \Theta and \|\cdot\|_F the Froebenius norm. Consider ... 1 vote 0 answers 1k views ### How to fill a rectangle with smaller rectangles of given sizes? I have a problem. I try to find an algorithm to fill up a given rectangle with smaller ones. Something like in this picture: I know the size of the big rectangle, the size of all the little ... 2 votes 1 answer 129 views ### Generating short Hamilton cycles from complete graphs Let G(V,E) be a complete symmetric graph without self-loops or parallel edges; depending on the context the edges may however be interpreted as a pair of antiparallel arcs of equal weight. A vertex ... 2 votes 1 answer 131 views ### Packing densities of non-centrally symmetric planar convex regions Reference: https://en.wikipedia.org/wiki/Smoothed_octagon Background: The smoothed octagon is conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex ... 1 vote 0 answers 210 views ### Why Densest packing of equal spheres in three dimensions is not 88.86? [closed] I placed four spheres of radius R at vertices of a tetrahedron of edge length 2R .When I calculated density I got 88.86.Actualy I wanted to calculate what is the maximum number of earth that can be ... 1 vote 0 answers 59 views ### A Variant of the Malfatti Problem See https://en.wikipedia.org/wiki/Malfatti_circles for an introduction to Malfatti's problem. The above page also states that for n >3, the question of whether a greedy method (at each step, the ... 1 vote 1 answer 133 views ### Effect of snowflaking on doubling constants This question is related to this one. Let (X,d) be a metric space, let \epsilon\in [0,1) and consider the snowflake (X,d^{1-\epsilon}). Suppose that (X,d) has a finite doubling constant, ... 1 vote 1 answer 76 views ### Worst convex compact set for translational packings of \mathbb R^d A (translational) packing of a convex compact subset (with non-empty interior) \mathcal C of \mathbb R^d is a union of translated non-overlapping (but perhaps touching) copies of \mathcal C. The ... 22 votes 0 answers 348 views ### What is the covering density of a very thin annulus? Is it \frac{\pi\sqrt{51\sqrt{17}-107}}{16}? Take some very small \epsilon>0, and consider the annulus/ring given by the set \{(r,\theta)\ |\ 1-\epsilon\le r\le1\}\subset \mathbb{R}^2. We wish to place translated copies of this annulus ... 1 vote 0 answers 303 views ### Which polygons tessellate the hyperbolic plane? The packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. It is well known that in Euclidean geometry, all triangles and all ... 4 votes 2 answers 258 views ### Which convex pentagon gives least packing density? Among all convex pentagons, does the regular pentagon give least packing density? Further question: For each n > 6, is the regular n-gon the minimum of packing density? An analogous question ... 8 votes 0 answers 351 views ### Covering number estimates for Hölder balls Let \alpha \in (0,1] and L>0. The Arzela-Ascoli Theorem guarantees that the set X(\alpha,L) of f:[-1,1]^n\rightarrow \mathbb{R}^m with \alpha-Hölder norm at-most L is compact in C([-1,... 9 votes 1 answer 191 views ### Which unimodular lattices L\subset \mathbb R^2 minimize f_t(L):=\sum_{ v\in L} e^{-t \|v\|_2}? (for parameters t>0) \DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}Consider the lattices in \SL(2,\mathbb R)(\mathbb Z^2) up to rotation. The space of such lattices can be identified with the modular surface \... 1 vote 3 answers 135 views ### On packing axisymmetric bodies in 3D Consider any 3D body with an axis of rotational symmetry (e.g. cone, cylinder...) and packing the 3d space efficiently with infinitely many copies of this body. Is the following claim valid? Claim: ... 3 votes 1 answer 96 views ### Covering radius of a lattice from relevant vectors Let L be an n-dimensional lattice. The Voronoi region of L is given by$$ \mathcal{V}(L)=\big\{x\in\mathbb{R}^n~|~ \|x\|_2\leq \|x-v\|_2~\forall v\in L\setminus\{0\}\big\}.  Considering the ...
I am looking for the sharpest known upper bound on $K(n, 1)$ as $n \rightarrow \infty$. This is the minimal cardinality of a (not-necessarily linear) covering code of $\{0, 1\}^n$ of radius 1. In ...
Suppose $Y$ is a random variable in $\mathbb{R}^d$, and we want to find the covering number \begin{equation*} \mathcal{F} = \big\{ F_{Y|W} (y | W) : y \in \mathbb{R}^d \big\} \end{equation*} where ...