Covering cross-polytopes with smaller homothetic copies

1.   Introduction

Let $K$ be a convex body in $\mathbb{R}^n$, i.e., a compact convex set having interior points. The set of convex bodies in $\mathbb{R}^n$ is denoted by $\mathcal{K}^n$, and the set of convex bodies that are centrally symmetric is denoted by $\mathcal{C}^n$. For each $x\in\mathbb{R}^n$ and each $\lambda > 0$, the set

is called a homothetic cop of $K$; when $\lambda\in(0, 1)$, it is called a smaller homothetic copy of $K$. For each $K\in\mathcal{K}^{n}$, we denote by $c(K)$ the least number of translates of ${\rm{int}} K$ needed to cover $K$. Concerning the least upper bound of $c(K)$ in $\mathcal{K}^{n}$, there is a long-standing conjecture (see ^{[1,2,3,4,5,6]} for the origin, history, and classical known results concerning this conjecture):

Conjecture 1. (Hadwiger's covering conjecture ^[4]) For each $K\in \mathcal{K}^{n}$, we have

and the equality holds if and only if $K$ is a parallelotope.

Although many people have conducted in-depth research, this conjecture is confirmed completely only for the planar case ^[7]. In ^[8], Chuanming Zong proposed a four-step program to attack this conjecture. In this program, it is important to estimate

i.e., $\Gamma_m(K)$ is the smallest positive number $\gamma$ such that $K$ can be covered by $m$ translates of $\gamma K$. The map $\Gamma_{m}(\cdot)$: $\mathcal{K}^n \to [0, 1]$, $K\mapsto \Gamma_{m}(K)$ is called the covering functional with respect to $m$, where $[m]: = \left\{{i\in\mathbb{Z}^{+}}\mid {1\le i\le m}\right\}$. Clearly, $c(K)\le m$ if and only if $\Gamma_{m}(K) < 1$. For each $m\in\mathbb{Z}^{+}$, $\Gamma_{m}(\cdot)$ is an affine invariant. More precisely,

where $\mathcal{A}^{n}$ is the set of non-degenerate affine transformations from $\mathbb{R}^n$ to $\mathbb{R}^n$.

A compact convex set $K$ is said to be an $d$-dimensional cross-polytope if there exist $d$ linearly independent vectors $v_1, \cdots, v_d$ such that

Clearly, any $d$-dimensional cross-polytope is the image of

under a non-degenerate affine transformation. Therefore, $\Gamma_{m}(K) = \Gamma_{m}(C_d)$ holds for each pair of positive integers $m$ and $d$. In a recent work ^[9], Xia Li et al. obtained some estimations of $\Gamma_{m}(C_d)$ for large $d$. Moreover, they showed that, if $P\in\mathcal{C}^{n}$ is a convex polytope with $2d$ vertices, then

which shows the importance of estimating $\Gamma_{m}(C_d)$.

It is well known that $\Gamma_{2^n}([-1, 1]^n) = 1/2, \; \forall n\geq 2$. It is interesting to ask whether there exists a universal upper bound for $\Gamma_{2^n}(C_n)$. In this paper, by using elementary yet interesting observations and refining techniques used in the recent works ^[9,10], we get better estimates of $\Gamma_{2^n}(C_n)$. Based on this, we present the first nontrivial universal upper bound of $\Gamma_{2^n}(C_n)$ for all positive $n$. By (1.1), results mentioned above yield also estimates of covering functionals of convex polytopes with few vertices.

Throughout this paper, the dimension $n$ of the underlying space is at least 3.

2.   Covering functionals of cross-polytopes

For each $n, k\in\mathbb{Z}^+$, we put

It is known that (cf. ^[11] or ^[12])

Lemma 1. Let $n, k\in\mathbb{Z}^+$. If $k\le \frac{n}{2}$, then

where

Moreover,

Proof. Let $(\alpha_1, \cdots, \alpha_n)$ be an arbitrary point in $(n+k)C_n$. Then

If $\sum_{i\in[n]}\left| {{\alpha _i}} \right|\le n$, then $(\alpha_1, \cdots, \alpha_n)\in nC_n\subseteq nC_n+S_k$. Otherwise, there exists $m\in[k]$ such that

On the one hand, since $\sum_{i\in[n]}(\left| {{\alpha _i}} \right|-\left \lfloor\left| {{\alpha _i}} \right| \right \rfloor) < n$, we have $\sum_{i\in[n]}\left \lfloor \left| {{\alpha _i}} \right| \right \rfloor\ge m$. Then there exist integers $\beta_1, \cdots, \beta_{n}\ge0$ such that

Clearly, we have

Therefore,

On the other hand, set

We have

By the Triangle Inequality, we have

Thus,

Without loss of generality, assume that $\alpha_1, \cdots, \alpha_n\ge 0$, and

By (2.1), we have

Then there exist integers $m_1', \cdots, m_n'$ such that

Set, for each $i\in[n]$, $f_i(\lambda) = |\alpha_i-\lambda|$. Then $f_i$ is decreasing on $[\beta_i, \lfloor \alpha_i \rfloor]$. We claim that

The case when $m_i'\in [\beta_i, \lfloor \alpha_i \rfloor]$ is clear. If $m_i' > \lfloor \alpha_i \rfloor$, then $m_i' = m_i = \lfloor \alpha_i \rfloor+1$ and $1/2\leq \alpha_i-\lfloor \alpha_i \rfloor < 1$. Thus

Hence (2.2) holds as claimed. It follows that

Therefore,

Moreover,

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For each $n\in\mathbb{Z}^+$, let $k_1(n)$ be the nonnegative integer satisfying

It is easy to prove that $k_1(n)\le \frac{n}{2}$.

Corollary 2. For each $n\in\mathbb{Z}^+$, we have

Remark 3. It can be verified that

For $x\in(0, +\infty)$, we define

Clearly, $g$ is strictly increasing on $(0, +\infty)$, and $\lim_{x\to0^+}g(x) = 1$. For each $t\in(1, +\infty)$, let $b(t)$ be the solution to the equation $g(x) = t$. Numerical calculation shows that $b(2)\approx 0.205597$. We can easily prove that, if $k_2(n)$ is the integer satisfying

then we have $\lim_{n\to\infty}\frac{k_2(n)}{n} = b(2)$ ^[9,10]. It can be verified that

Therefore, the estimate in Corollary 2 is slightly better than that given by ^[9, Proposition 5] in the asymptotical sense, and it is much better for particular choices of small $n$. For example, we have $k_1(7) = 2$ and $k_2(7) = 1$. It follows that

which is better than $\Gamma_{128}(C_7)\le\frac{n}{n+k_2(n)} = \frac{7}{7+1}\le0.875$ ^[9]. See Table 1 for more examples.

Theorem 4. For each $n\ge 3$, we have

Proof. By numerical calculations, for each $3\le n\le 49$, we have $\Gamma_{2^n}(C_n)\le \frac{6}{7}$. Set $c = b(2)-0.02$. Then for each $n\ge 50$, we have $cn\le b(2)n-1$, which shows that $(1+c)n\le n+\left \lfloor b(2)n \right \rfloor$. Therefore,

Thus, for each $n\ge 3$, we have

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3.   Conclusions

By refining techniques used in the recent works ^[9,10], we get better estimates of $\Gamma_{2^n}(C_n)$ and the first nontrivial universal upper bound of $\Gamma_{2^n}(C_n)$. It is natural to find universal bounds of $\Gamma_{2^n}(B_p^n)$ for fixed $p\in (1, \infty)$, where $B_p^n$ is the closed unit ball of $(\mathbb{R}^n, \left\| \cdot \right\|_p)$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are supported by the [National Natural Science Foundation of China] grant numbers [12071444] and the [Fundamental Research Program of Shanxi Province of China] grant numbers [202103021223191].

Conflict of interest

There is no conflicts of interest between all authors.