On weighted covering numbers and the Levi-Hadwiger conjecture

Covering numbers can be found in various fields of mathematics, including combinatorics, probability, analysis and geometry. They often participate in the solution of many problems in quite a natural manner.

In the combinatorial world, the idea of fractional covering numbers is well-known and utilized for many years. In [2], the authors introduced the weighted notions of covering and separation numbers of convex bodies and shed new light on the relations between the classical notions of covering and separation, as well as on the relations between the classical and weighted notions. In this note we propose a variant of these numbers which is perhaps more natural and discuss these numbers in more detail, revealing more useful relations, as well as some applications. To state our main results, we need some definitions. The impatient reader may skip the following section and go directly to Section 1.3 where the main results are stated.

Apart from deepening our understanding of these notions, and revealing more useful relations, we also consider this work as a first step towards the functionalization of covering and separation numbers; in the past decade, various parts from the theory of convex geometry have been gradually extended to the realm of log-concave functions. Numerous results found their functional generalizations. One natural way to embed convex sets in $\mathbb{R}^{n}$ into the class of log-concave functions is to identify every convex set $K$ with its characteristic function $\mathbbm{1}_{K}$. Besides being independently interesting, such extensions may sometimes be applied back to the setting of convex bodies. For further reading, we refer the reader to [1, 3, 4, 14, 15]. Since covering numbers play a considerable part in the theory of convex geometry, their extension to the realm of log-concave functions seems to be an essential building block for this theory. Our results using functional covering numbers will be published elsewhere.

Let $K\subseteq\mathbb{R}^{n}$ be compact and let $T\subseteq\mathbb{R}^{n}$ be compact with non-empty interior. The classical covering number of $K$ by $T$ is defined to be the minimal number of translates of $T$ such that their union covers $K$, namely

$$N\left(K,T\right)=\min\left\{N\,:\,N\in\mathbb{N},\,\,\exists x_{1},\dots x_{N}\in\mathbb{R}^{n};\,\,K\subseteq\bigcup_{i=1}^{N}\left(x_{i}+T\right)\right\}.$$

Here and in the sequel we assume that the covered set $K$ is compact and the covering set $T$ has non-empty interior so that the covering number will be finite. However, one may remove these restriction so long as we are content also with infinite outcomes.

A well-known variant of the covering number is obtained by considering only translates of $T$ that are centered in $K$, namely

$$\overline{N}\left(K,T\right)=\min\left\{N\,:\,N\in\mathbb{N},\,\,\exists x_{1},\dots x_{N}\in K;\,\,K\subseteq\bigcup_{i=1}^{N}\left(x_{i}+T\right)\right\}.$$

Clearly, $N\left(K,T\right)\leq\overline{N}\left(K,T\right)$, and it is easy to check that for convex bodies¹¹1by convex body we mean, here and in the sequel, a compact convex set with non-empty interior $K$ and $T$, we have $\overline{N}\left(K,T-T\right)\leq N\left(K,T\right)$. Furthermore, if $T$ is a Euclidean ball then $N\left(K,T\right)=\overline{N}\left(K,T\right)$.

The classical notion of the separation number of $T$ in $K$ is closely related to covering numbers and is defined to be the maximal number of non-overlapping translates of $T$ which are centered in $K$;

$$M\left(K,T\right)=\max\left\{M\,:\,N\in\mathbb{N},\,\,\exists x_{1},\dots x_{M}\in K\,;\,\,\left(x_{i}+T\right)\cap\left(x_{j}+T\right)=\emptyset\,\,\forall i\neq j\right\}.$$

It is a standard equivalence relation that $N\left(K,T-T\right)\leq M\left(K,T\right)\leq N\left(K,T\right)$. We also define the less conventional

$$\overline{M}\left(K,T\right)=\max\left\{M\,:\,N\in\mathbb{N},\,\,\exists x_{1},\dots x_{M}\in K\,;\,\,\left(x_{i}+T\right)\cap\left(x_{j}+T\right)\cap K=\emptyset\,\,\forall i\neq j\right\}.$$

Note that the condition $\left(x_{i}+T\right)\cap\left(x_{j}+T\right)=\emptyset$ is equivalent to $x_{i}-x_{j}\not\in T-T$ which means that $M\left(K,T\right)=M\left(K,-T\right)=M\left(K,\frac{T-T}{2}\right)$. Moreover, it is easily checked that for a convex $K$ and for a centrally symmetric convex body $L$ (i.e., $L=-L$) we have $M\left(K,L\right)=\overline{M}\left(K,L\right)$ and thus by the last remark $M\left(K,T\right)=\overline{M}\left(K,T\right)$ for any convex body $T$. In the sequel, we will define weighted counterparts for $M\left(K,T\right)$ and $\overline{M}\left(K,T\right)$ which will not necessarily be equal, even in the convex and centrally symmetric case.

In order to define the weighted versions, let $\mathbbm{1}_{A}$ denote the indicator function of a set$A\subseteq\mathbb{R}^{n}$, equal to $1$ if $x\in A$ and $0$ if $x\not\in A$.

Let us reformulate the above definitions in the language of measures. Note that the covering condition $\sum_{i=1}^{N}\omega_{i}\mathbbm{1}_{x_{i}+T}\left(x\right)\geq 1$ for all $x\in K$ is equivalent to $\nu*\mathbbm{1}_{T}\geq\mathbbm{1}_{K}$ where $\nu=\sum_{i=1}^{N}\omega_{i}\delta_{x_{i}}$ is the discrete measure with masses $\omega_{i}$ centered at $x_{i}$ and where $*$ stands for the convolution

$$\left(\nu*\mathbbm{1}_{T}\right)\left(x\right)=\int_{\mathbb{R}^{n}}\mathbbm{1}_{T}\left(x-y\right)d\nu\left(y\right).$$

Let ${\cal D}_{+}^{n}$ denote all non-negative discrete and finite measures on $\mathbb{R}^{n}$ and let ${\rm supp}\left(\nu\right)\subseteq\mathbb{R}^{n}$ denote the support of a measure $\nu$ on $\mathbb{R}^{n}$. Thus, the weighted covering numbers of $K$ by $T$ can be written as

$$N_{\omega}\left(K,T\right)=\inf\left\{\nu(\mathbb{R}^{n})\,:\,\nu*\mathbbm{1}_{T}\geq\mathbbm{1}_{K}\,,\nu\in{\cal D}_{+}^{n}\right\}$$

and

$$\overline{N}_{\omega}\left(K,T\right)=\inf\left\{\nu(\mathbb{R}^{n})\,:\,\nu*\mathbbm{1}_{T}\geq\mathbbm{1}_{K}\,,\nu\in{\cal D}_{+}^{n}\,\,{\rm with}\,\,{\rm supp}\left(\nu\right)\subseteq K\right\}.$$

It is this natural to extend this notion of covering to general non-negative measures. Let ${\cal B}_{+}^{n}$ denote all non-negative Borel regular measures on $\mathbb{R}^{n}$.

The weighted notions of the separation are defined similarly; a measure $\mu\in{\cal B}_{+}^{n}$ is said to be $T-$separated if $\mu*\mathbbm{1}_{T}\leq 1$. The weighted separation numbers, corresponding to $N_{\omega}\left(K,T\right)$, $\overline{N}_{\omega}\left(K,T\right)$ and $N^{*}\left(K,T\right)$ are respectively defined by:

$$M_{\omega}\left(K,T\right)=\sup\left\{\int_{K}d\nu\,:\,\nu*\mathbbm{1}_{T}\leq 1,\,\,\nu\in{\cal D}_{+}^{n}\right\},$$

$$\overline{M}_{\omega}\left(K,T\right)=\sup\left\{\int_{K}d\nu\,:\,\,\forall x\in K\,\,\left(\nu*\mathbbm{1}_{T}\right)\left(x\right)\leq 1,\,\,\nu\in{\cal D}_{+}^{n}\right\}$$

and

$$M^{*}\left(K,T\right)=\sup\left\{\int_{K}d\mu\,:\,\mu*\mathbbm{1}_{T}\leq 1\,,\mu\in{\cal B}_{+}^{n}\right\},$$

where again clearly $M_{\omega}\left(K,T\right)\leq M^{*}\left(K,T\right)$.

Our first main result is a strong duality between weighted covering and separation numbers; it turns out that $N^{*}\left(K,T\right)$ and $M^{*}\left(K,-T\right)$ can be interpreted as the outcome of two dual problems in the sense of linear programming. Indeed, as in [2], this observation is a key ingredient in the proof of our first main result below which states that the outcome of these dual problems is the same (we call this “strong duality”).

As a consequence of Theorem 1.3, together with the well-known homothety equivalence between classical covering and separation numbers $N\left(K,T-T\right)\leq M\left(K,T\right)\leq N\left(K,T\right)$, we immediately get the following equivalence relation between the classical and weighted covering numbers (which has also appeared in [2] for the pair $\overline{M}_{\omega},\overline{N}_{\omega}$).

For a centrally symmetric convex set $T$, Corollary 1.5 reads $N\left(K,2T\right)\leq N_{\omega}\left(K,T\right)\leq N\left(K,T\right)$. Although this “constant homothety” equivalence of classical and weighted covering is useful, it turns out to be insufficient in certain situations. To that end, we introduce our second main result, in which the homothety factor $2$ is replaced by a factor $1+\delta$ with $\delta>0$ arbitrarily close to $0$. This gain is diminished by an additional logarithmic factor; such a result is a reminiscent of Lovï¿œsz’s [17] well-known inequality for fractional covering numbers of hypergraphs.

We remark that for the proof of our application in Section 3 below, we shall use $T_{1}=\delta T$ and $T_{2}=\left(1-\delta\right)T$ for $0<\delta<1$ and a single convex body $T$. It is also worth mentioning that Theorem 1.6 holds for $\overline{N}_{\omega}\left(K,T\right)$ and $\overline{N}\left(K,T\right)$ as well (with the exact same proof).

Let ${\rm Vol}\left(A\right)$ denote the Lebesgue volume of a set $A\subseteq\mathbb{R}^{n}$. The classical covering and separation numbers satisfy simple volume bounds. Such volume bounds also hold for the weighted case, and turn out to be quite useful.

A famous conjecture, known as the Levi-Hadwiger or the Gohberg-Markus covering problem, was posed in [16], [13] and [12]. It states that in order to cover a convex set by slightly smaller copies of itself, one needs at most $2^{n}$ copies.

This problem has drawn much attention over the years, but only little has been unraveled so far. We mention that Levi confirmed the conjecture for the plane, and that Lassak confirmed it for centrally symmetric bodies in $\mathbb{R}^{3}$. The currently best known general upper bound for $n\geq 3$ is ${2n\choose n}\left(n\ln n+n\ln\ln n+5n\right)$ and the best bound for centrally symmetric convex bodies is $2^{n}$$\left(n\ln n+n\ln\ln n+5n\right)$, both of which are simple consequences of Rogers’ bound for the asymptotic lower densities for covering the whole space by translates of a general convex body, see [20]. For a comprehensive survey of this problem and the aforementioned results see [8].

It is natural, after introducing weighted covering, to formulate the Levi-Hadwiger covering problem for the case of weighted covering.

For centrally symmetric convex bodies, we verify Conjecture 1.9, including the equality case. We show

It is worth mentioning that the classical covering problem of Levi-Hadwiger is equivalent to the problem of the illumination of a convex body (for surveys see [18, 6]) which asks how many directions are required to illuminate the entire boundary of a convex body $K$ (a direction $u\in\mathbb{S}^{n-1}$ is said to illuminate a point $b$ in the boundary of $K$ if the ray emanating from $b$ in direction $u$ intersects the interior of $K$). A fractional version of the illumination problem was considered in [19], where it was proven that the fractional illumination number of a convex body $K$, denoted by $i^{*}\left(K\right)$, satisfies that $i^{*}\left(K\right)\leq{2n\choose n}$ and that $i^{*}\left(K\right)\leq 2^{n}$ for all centrally symmetric bodies (with parallelotopes attaining equality). It was further conjectured [19, Conjecture 7] that $i^{*}\left(K\right)\leq 2^{n}$ for all convex bodies and that equality is attained only for parallelotopes. However, as no relation between fractional and usual illumination numbers was proposed, this result remained isolated. Also, it seems that the equality conditions were not analyzed. In fact, one may verify that the proof of the equivalence between the illumination problem and the Levi-Hadwiger covering problem (see [7, Theorem 7]) carries over to the fractional setting and conclude that $i^{*}\left(K\right)=\lim_{\lambda\to 1^{-}}N_{\omega}\left(K,\lambda K\right)$. Thus, Theorem 1.10 actually verifies the aforementioned results about fractional illumination and also verifies [19, Conjecture 7] for the case of centrally symmetric convex bodies, including the equality hypothesis.

Combining the inequality in Theorem 1.6 with the volume inequality in Theorem 1.7, we prove the following bound for the classical Levi-Hadwiger problem, in the case of centrally symmetric convex bodies, which is the same as the aforementioned (best known) general bound of Rogers.

We thank Prof. Noga Alon, Prof. Mark Meckes and Prof. Boris Tsirelson for their valuable comments and suggestions. We also thank Prof. Rolf Schneider for his proof of Lemma 3.1 and for translating for us the entire paper [9] from German.
This research was supported in part by ISF grant number 247/11.

The remainder of this note is organized as follows. In Section 2.1 we show weak duality between weighted covering and separation numbers. In Section 2.2 we prove Theorem 1.3. In Section 2.3 we discuss the existence of optimal covering measures. In Section 2.4 we discuss the approximation of uniform covering measures by discrete covering measures. In 2.5 we prove Theorem 1.7. In Section 2.6 we prove Theorem 1.6. In Section 2.7 we discuss the weighted notions of covering and separation in the setting of general metric spaces. In Section 3 we discuss both the classical and weighted versions of the Levi-Hadwiger covering problems, proving Theorem 1.10 and Corollary 1.11.

In this section we prove Theorem 1.3. By Proposition 2.1 it is enough to show an inequality $M_{\omega}\left(K,T\right)\geq N_{\omega}\left(K,-T\right)$.

We start with the discretized versions of our weighted covering and separation notions. Let $\Lambda=\left\{x_{i}\right\}_{i=1}^{d}\subseteq\mathbb{R}^{n}$ be some finite set, which will be chosen later, and define:

$$N_{\omega}\left(K,T,\Lambda\right)=\inf\left\{\sum_{i=1}^{N}\omega_{i}\,:\,\,\exists\left(x_{i},\,\omega_{i}\right)_{i=1}^{N}\subseteq\left(\Lambda,\,\mathbb{R}^{+}\right),\,\,\sum_{i=1}^{N}\omega_{i}\mathbbm{1}_{T}\left(x-x_{i}\right)\geq 1_{K}(x)\,\>\forall x\in\Lambda\right\}$$

and

$$M_{\omega}\left(K,T,\Lambda\right)=\sup\left\{\sum_{i=1}^{N}\omega_{i}\,:\,\,\exists\left(x_{i},\,\omega_{i}\right)_{i=1}^{N}\subseteq\left(\Lambda\cap K,\,\mathbb{R}^{+}\right),\,\,\sum_{i=1}^{N}\omega_{i}\mathbbm{1}_{T}\left(x-x_{i}\right)\leq 1\,\>\forall x\in\Lambda\right\}.$$

In this setting, linear programming duality gives us an equality of the form

$$N_{\omega}\left(K,T,\Lambda\right)=M_{\omega}\left(K,-T,\Lambda\right).$$

(2.1)

Indeed, define the vectors $b,c\in\mathbb{R}^{d}$ by

$$c_{i}=\begin{cases}1,&x_{i}\in K\\ 0,&{\rm otherwise}\end{cases}\,\,,\,\,\,\,\,\,\,\,b_{i}=1$$

and the $d\times d$ matrix $M$ by

$$M_{ij}=\begin{cases}1,&x_{i}\in x_{j}+T\\ 0,&{\rm otherwise.}\end{cases}$$

Note that

$$M_{ij}^{T}=\begin{cases}1,&x_{i}\in x_{j}-T\\ 0,&{\rm otherwise.}\end{cases}$$

Let $\langle\cdot,\,\cdot\rangle$ denote the standard Euclidean inner product in $\mathbb{R}^{d}$. Then, in the language of vectors and matrices, the above discretized weighted covering and separation notions read

	$\displaystyle N_{\omega}\left(K,T,\Lambda\right)$	$\displaystyle=$	$\displaystyle\min\left\{\langle b,\,x\rangle:\,Mx\geq c,\,x\geq 0\right\},$
	$\displaystyle M_{\omega}\left(K,-T,\Lambda\right)$	$\displaystyle=$	$\displaystyle\max\left\{\langle c,\,y\rangle:\,M^{T}y\leq b,\,y\geq 0\right\}$

which are equal by the well-known duality theorem of linear programming, see e.g., [5].

Next, we shall use this observation with a specific family of sets $\Lambda(\delta).$ A set $\Lambda\left(\delta\right)\subseteq\mathbb{R}^{n}$ is said to be a $\delta-$net of a set $A\subseteq\mathbb{R}^{n}$ if for every $x\in A$ there exists $y\in\Lambda\left(\delta\right)$ for which $\left|x-y\right|\leq\delta$. In other words, $A\subseteq\Lambda+\delta B_{2}^{n}$. We shall make use of the two following simple lemmas, corresponding to [2, Lemmas 14-15].

Finally, to prove Theorem 1.3 we shall need the following continuity result for weighted covering numbers:

In this section our goal is somewhat technical. We wish to use a uniform measure to bound $N_{\omega}(K,T)$, however it is not a member of ${\cal D}_{+}^{n}$. We claim that if we find some uniform covering measure of a set $K$ by a convex set $T$ (supported on some compact Borel set) with total mass $m$, then $N_{\omega}(K,T)\leq m$. This is because uniform measures can be approximated well by discrete ones, and requires a proof. To this end, we need to recall the definition of a Glivenko-Cantelli class. Let $\xi_{1},\xi_{2},\dots$ be a sequence of i.i.d $\mathbb{R}^{n}$-valued random vectors having common distribution $P$. The empirical measure $P_{k}$ is formed by placing mass $1/k$ at each of the points $\xi_{1}\left(\omega\right),\xi_{2}\left(\omega\right),\dots,\xi_{k}\left(\omega\right)$ . A class ${\cal A}$ of Borel subsets $A\in{\cal A}$ of $\mathbb{R}^{n}$ is said to be a Glivenko-Cantelli class for $P$ if

$$\sup_{A{\cal\in A}}\left|P_{n}\left(A\right)-P\left(A\right)\right|\overset{a.s.}{\longrightarrow}0$$

In the following lemma, we will invoke a Glivenko-Cantelli theorem for the class ${\cal C}_{n}$ of convex subsets of $\mathbb{R}^{n}$. Namely, in [10, Example 14] it is shown that if a probability distribution $P$ satisfies that $P(\partial K)=0$ for all $K\in{\cal C}_{n}$ then ${\cal C}_{n}$ is a Glivenko-Cantelli class for $P$.