# 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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### How many Hamming spheres of radius 1 does it take to cover the cube?

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 ...
• 61
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 ...