# Questions tagged [packing-and-covering]

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### An arrangement of entangled squares

Is there an arrangement of finitely many axes-parallel squares in the plane, of $k$ different colors, such that: The squares of each color are pairwise-disjoint; Each square overlaps at least $4$ ...
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### Separation of balls in the torus

Let $X_1, \dots, X_N$ be $N$ balls of radius $R<<1$ in $[0,1]^d$ such that $N R^d \leqslant R^{\alpha}$ for some $\alpha > 0$ and $d(x_i,x_j)\geqslant 2R$ for any $i\not=j$. The assumption ...
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### Packing a Riemannian manifold with disjoints balls

Let $M$ be a smooth Riemannian manifold with riemannian measure $\mu$. I don't suppose that $M$ is complete. Can we find a finite or countable disjoint collection of open (or closed) and relatively ...
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### Covering number of $l_2$ Ball in $\mathbb{R}^d$

What is the covering number $N(\epsilon, B_2, ||\cdot||_2)$ of a ball $B_2$ in $\mathbb{R}^d$ of radius $r$ under the $l_2$ norm?
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### Stronger version of Besicovitch covering theorem

I'm wondering if the following strengthening of the Besicovitch covering theorem holds: Suppose $A\subset\mathbb R^n$ is a bounded subset and suppose $x\mapsto r_x$ is a function $A\to(0,\infty)$. Is ...
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### Is there a simple algorithm to fit rectangles of differing sizes into a larger rectangular-container through one side

Imagine a shipping container (in the two-dimensional length-width dimension). Let's further assume there is a queue of parts waiting to be fit into this container. This queue is not infinite but ...
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### Asymptotic bound on minimum epsilon cover of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; ...
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### Subdividing a Compact Bounded Curvature Manifold into Charts with Bounded Lipschitz Constant

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$ cover $P$ of $M$; that is ...
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### Uniform inequality of the form $\text{Proba}(\sup_{v \in [-M,M]^k}|p^Tv-\hat{p}_n^Tv| \le \epsilon_n) \ge 1 - \delta$

Let $M > 0$, $k$ be a positive integer, and $\mathcal V:=[-M,M]^k$. Finally, let $p \in \Delta_k$, (where $\Delta_k$ is the $(k-1)$-dimensional probability simplex) and let $\hat{p}_n$ be an ...
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### Covering a simplex efficiently by efficiently describable polytopes?

Take a standard simplex or cube in $\mathbb R^n$. Is there a way to cover it with $O(poly(\log n))$ convex polytopes each describable by only $O(poly(\log n))$ half-plane inequalities? If not what ...
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### Polytopes that can be efficiently described and efficiently covered by cubes or simplices?

Is there a bounded convex polytope $\mathcal P\subseteq\mathbb R^n$ with $m$ vertices, whose vertex vectors span $\mathbb R^n$ (so $m$ is $\Omega(n)$) and just $O(poly(\log n))$ half-plane ...
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### When can any graph $G$ be expressed as a union of $\alpha(G)$ complete graphs?

If for any graph $H$ we define $\alpha(H)$ to be the cardinality of any maximum size indepedent set in $H$. Then under what conditions can any graph $G$ be expressed as a union of $\alpha(G)$ complete ...
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### Fitting $\frac1n\times\frac1{n+1}$ rectangles into the unit square [duplicate]

Consider the set of rectangles $r_n | n \in \Bbb N$ such that rectangle $r_n$ has shape $\frac1n\times\frac1{n+1}$. The total area composed by one copy of each $r_n$ as $n$ ranges from $1$ to ...
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### Can 4-space be partitioned into Klein bottles?

It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles, or into disjoint unit circles, or into congruent copies of a real-analytic curve (Is it possible to partition $\mathbb R^3$ ...
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### Lower bound on misclassification rate of Lipschitz functions in terms of Lipschitz constant

Important note @MateuszKwaśnicki in the comment section has raised a fundamental issue with the current statement of the problem. I'm trying to bugfix it. Setup I wish to show that a Lipschitz ...
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### VC dimension, fat-shattering dimension, and other complexity measures, of a class BV functions

I wish to show that a function which is "essentially constant" (defined shortly) can't be a good classifier (machine learning). For this i need to estimate the "complexity" of such a class of ...
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### Which finite sets could be packed into a square?

This question is inspired by an interesting visualization of the finite levels of von Neumann's hierarchy on Adam P. Goucher's blog, Complex Projective 4-Space. The problem starts with a two-...
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### Counting triples family with double shared elements

We have a set of elements $S=\{1,2,3, . . . , N\}$ and family $F$ of $N$ triples of elements in $S$ ($N$ is a multiple of 3). Each element of $S$ appears in exactly three triples. The elements in each ...
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### Packing number of Lipschitz functions

For some $L>0$ say ${\cal L}$ is the space of all $L-$Lipschitz functions mapping $(X,\rho) \rightarrow [0,1]$ where $(X,\rho)$ is a metric space. For any $\alpha >0$ do we know of a ...
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### Packing points in a lattice

Let $L$ be the square or triangular lattice in the plane, with nearest neighbors having distance 1. Has anyone studied the problem of finding the maximum (okay, supremum) density achieved by a subset ...
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### Sphere packings with antipodal (unequal) spheres

Let $\|\cdot\|_2$ denote the Euclidean norm, let $\langle \cdot, \cdot\rangle$ denote the standard dot product, and let $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ ...
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### Packing net of simplex

For given $d$, we can define the simplex as follows, $S=\{(x_1,x_2,\cdots,x_d):x_1\geq x_2\geq \cdots\geq x_d\geq 0,\sum x_i=1\}$. We can define the distance on $S$ as $L_1$ distance. An $\epsilon$ ...
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### Thinnest covering of the plane by regular pentagons

Q. Is it known what is the thinnest covering of the infinite plane by regular pentagons? By covering I mean every point of the plane is covered. By thinnest I mean the proportion of the plane covered ...
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### Equation of state for hard rods

Some context: For ideal gases, the thermodynamic equation of state is the well-known: $$pV = nRT \tag{1}$$ where $n$ is the amount of substance, $R$ the universal gas constant and $P,V,T$ are ...
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### Sequential addition of points on a circle, optimizing asymptotic packing radius

Suppose I have to put $N$ points $x_1, x_2, \ldots, x_N$ on the circle $S^1$ of length 1 so as to achieve the largest minimum separation (packing radius). The optimal solution is the equally spaced ...
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### Packing the box $[0,X^{\delta_1}] \times [0,X^{\delta_2}] \times [0,X^{\delta_3}] \subseteq \mathbb{R}^3$ with cubes

Let $0 < \delta_1 \leq \delta_2 \leq \delta_3 \leq 1$, and consider the box $B := [0,X^{\delta_1}] \times [0,X^{\delta_2}] \times [0,X^{\delta_3}] \subseteq \mathbb{R}^3$. Let $X > 3$ say. Is it ...
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### Packing of anisotropic objects

The most famous version of packing problems deals with perfectly symmetrical shapes such as spheres. But how about anisotropic shapes? More prcisely, if we want to compare spherocylinders (cylinders ...
Given $n$ and $B_n= \lceil H_n \rceil$, where the latter is the $n$th harmonic number $\sum^n_{i=1} 1/i$, for most $n$ it is easy to pack the first $n$ terms of the harmonic series into $B_n$ many ...
Given a polygon and dimension $d$, find a minimum partition of rectangles that has either of its dimensions equal to $d$. Example: Consider the following diagram: I want to cover maximum shaded ...