1 Introduction
Let
${\mathcal {K}}_n$
be the family of all convex bodies in
${\mathbb {R}}^n$
, i.e., all convex compact sets
$K\subset {\mathbb {R}}^n$
with a nonempty interior (
$\textrm {int}(K)\neq \emptyset $
). For
$A,B\subset {\mathbb {R}}^n$
, we denote by
$$\begin{align*}C(A,B):=\min \left\{N:\exists t_1,\dots,t_N\in{\mathbb{R}}^n \text{ satisfying } A\subseteq \bigcup_{j=1}^N(t_j+B)\right\}, \end{align*}$$
the minimal number of translates of B needed to cover A.
Hadwiger [Reference Hadwiger18] raised the question of determining the value of
$$\begin{align*}H_n=\max \{C(K,\textrm{int}(K)):K\in{\mathcal{K}}_n\} \end{align*}$$
for all
$n\ge 3$
.
Considering an n-cube, one immediately sees that
$H_n\ge 2^n$
, and the well-known Hadwiger’s covering conjecture states that
$H_n=2^n$
for
$n\ge 3$
with equality only for parallelepipeds. It was shown that
$H_2=4$
by Levi [Reference Levi24]. Aside from Levi and Hadwiger, the conjecture may be associated with the names of Boltyanski, who in [Reference Boltyanski8] has established an equivalent formulation in terms of illumination of the boundary of the body by external light sources, and with Gohberg–Markus [Reference Gohberg and Markus16] who asked the question in terms of the minimal number of smaller homothetic copies of K required to cover K. As of today, the conjecture, which is also known as the illumination conjecture, is wide open. For details about the history and partial results for special classes of convex bodies see, e.g., [Reference Bezdek and Khan6]. For general background on convex geometry one can refer to, e.g., [Reference Schneider33] or to the introductory sections of [Reference Artstein-Avidan, Giannopoulos and Milman2] or [Reference Brazitikos, Giannopoulos, Valettas and Vritsiou9].
In what follows, we will outline the current approaches in obtaining the upper bounds on
$H_n$
and our modifications allowing to obtain new bounds.
To this end, let
${\mathcal {K}}_n^s$
be the subfamily of
${\mathcal {K}}_n$
consisting of centrally symmetric convex bodies, and define
$$ \begin{align*}H_n^s=\max \{C(K,\textrm{int}(K)):K\in{\mathcal{K}}_n^s\}.\end{align*} $$
The best known explicit upper bound on
$H_n$
in high dimensions follows from the results of Rogers and Zong [Reference Rogers and Zong30] (which, in turn, use the Rogers–Shephard inequality [Reference Rogers and Shephard29]):
$$ \begin{align} H_n\le \binom{2n}{n}n(\ln n+\ln\ln n+5). \end{align} $$
This bound is valid for each dimension
$n\ge 3$
, so it can be used for our settings of specific small n. We remark that a similar estimate was first obtained by Erdös and Rogers [Reference Erdős and Rogers13], who showed (1.1) with
$4+\tfrac 1n$
in place of
$5$
, but only for sufficiently large n. Now the asymptotic behavior of
$\binom {2n}{n}$
is
$\frac {4^n}{\sqrt {2\pi n}}$
, so this upper bound is
$(4+o(1))^n$
while the conjecture states
$H_{n}=2^n$
. Remarkable sub-exponential improvements of (1.1) have been obtained only recently by Huang, Slomka, Tkocz, and Vritsiou [Reference Huang, Slomka, Tkocz and Vritsiou19]. Namely, using “thin-shell” volume estimates they obtained
$$ \begin{align} H_n\le \exp(-c\sqrt{n}) 4^n. \end{align} $$
This was later improved, using the new breakthrough bounds on the isotropic constant, by Campos, van Hintum, Morris, and Tiba [Reference Campos, van Hintum, Morris and Tiba10] to
$$ \begin{align} H_n\le \exp\Bigl(\frac{-cn}{\ln n}\Bigr)4^n. \end{align} $$
Both constants c in (1.2) and (1.3) are independent of n but not given explicitly. For centrally symmetric bodies, one has (see [Reference Erdős and Rogers13] and [Reference Rogers and Zong30])
$$ \begin{align} H_n^s\le 2^n n(\ln n+\ln\ln n+5), \end{align} $$
which is asymptotically close to the conjectured
$2^n$
.
The first step in obtaining any of the inequalities (1.1)–(1.4) is the result of Rogers [Reference Rogers28] on the covering density of
${\mathbb {R}}^n$
by translates of a body L. Namely, let
$$ \begin{align*} \theta(L)=\lim_{R\to\infty}\inf_{\Lambda}\left\{ \frac{|L|\#\Lambda}{(2R)^n}: [-R,R]^n\subseteq \bigcup_{\lambda\in\Lambda} (x_\lambda+L) \right\} \end{align*} $$
denote the covering density of space by translates of L, where
$|L|$
is the volume of L, and
$\#\Lambda $
is the cardinality of
$\Lambda $
. Rogers [Reference Rogers28] showed that for any body
$L\in {\mathcal {K}}_n$
,
$$ \begin{align} \theta(L)\le n(\ln n+\ln\ln n+5). \end{align} $$
The main idea of [Reference Rogers28] is that an appropriate number of random translates of L covers “most” of the space, and the leftover can be handled using a maximum packing argument.
The second step is the use of the inequality
$$ \begin{align} C(K,L)\le \frac{|K-L|}{|L|}\theta(L), \end{align} $$
which is valid for any
$K,L\in {\mathcal {K}}^n$
and was proved by Rogers and Zong [Reference Rogers and Zong30] using certain averaging argument. (Here
$K-L=\{x-y:x\in K,y\in L\}$
is the Minkowski difference.)
For symmetric bodies, one obtains (1.4) from (1.6) by choosing
$L=(1-\varepsilon )K$
with
$\varepsilon \to 0+$
and using the fact that
$|K-K|=|2K|=2^n|K|$
for
$K\in {\mathcal {K}}_n^s$
. For general bodies, (1.1) follows from (1.6) by applying the Rogers–Shephard inequality (see [Reference Rogers and Shephard29]), which, for any
$K\in {\mathcal {K}}_n$
, estimates the volume of the difference body
$K-K$
by
$|K-K|\le \binom {2n}{n}|K|$
.
The improvements (1.2) and (1.3) are both based on choosing L to be the largest (by volume) centrally symmetric subset of K. With such choice, we have
$|K-L|/|L|\le |2K|/|L|\le 2^n \Delta _{KB}(K)^{-1}$
, where
$$\begin{align*}\Delta_{KB}(K):=\max_{x\in {\mathbb{R}}^n}\frac{|K\cap (x-K)|}{|K|} \end{align*}$$
is the Kövner–Besicovitch measure of symmetry of K. Simple averaging gives
$\Delta _{KB}(K)\ge 2^{-n}$
for any
$K\in {\mathcal {K}}_n$
, while [Reference Huang, Slomka, Tkocz and Vritsiou19] and [Reference Campos, van Hintum, Morris and Tiba10] obtain the corresponding sub-exponential improvements to the lower bound on
$\Delta _{KB}(K)$
, yielding (1.2) and (1.3).
For low dimensions, Lassak [Reference Lassak22] showed that
$$ \begin{align} H_n\le (n+1)n^{n-1}-(n-1)(n-2)^{n-1}, \end{align} $$
which outperforms (1.1) for
$n\le 5$
. For
$n=3$
this gives
$H_3\le 34$
, which was improved to
$H_3\le 20$
by Lassak [Reference Lassak23], then to
$H_3\le 16$
by Papadoperakis [Reference Papadoperakis25], and then to
$H_3\le 14$
by Prymak [Reference Prymak26]. For slightly larger dimensions, it was shown in [Reference Prymak and Shepelska27] that
$H_4\le 96$
,
$H_5\le 1091$
and
$H_6\le 15373$
improving both (1.1) and (1.7) for
$n=4,5,6$
. Then Diao[Reference Diao12] obtained
$H_5\le 1002$
and
$H_6\le 14140$
. All of these results in low dimensions were based on comparing the body with a suitable parallelepiped. For the symmetric case, Lassak [Reference Lassak21] obtained the sharp result
$H_3^s=8$
, but for
$n\ge 4$
no estimate better than (1.4) or than the corresponding bound on
$H_n$
(obviously,
$H_n^s\le H_n$
) was known.
Our main result is the following new bounds.
Theorem 1.1
$H_5\le 933$
,
$H_6\le 6137$
,
$H_7\le 41377$
,
$H_8\le 284096$
,
$H_4^s\le 72$
,
$H_5^s\le 305$
, and
$H_6^s\le 1292$
.
The main idea of the proof is to utilize (1.6) with L being the maximal (by volume) inscribed ellipsoid into K (such ellipsoids were characterized by John [Reference John20]). After an appropriate affine transform, we can assume that L is the unit ball
$B_2^n$
in
${\mathbb {R}}^n$
, while K is in the so-called John’s position, so applying (1.6) we get
$$ \begin{align}C(K, \textrm{int}{K})\leq \frac{|K+B_2^n|}{|B_2^n|}\theta(B_2^n).\end{align} $$
Next we make use of several geometric results which allow us to obtain an upper bound on
$|K+B_2^n|$
. One ingredient in such estimates is the fact that the mean width and the volume of a body in John’s position are largest for the regular simplex (general case) or for the cube (symmetric case), the results due to Ball [Reference Ball3], Barthe [Reference Barthe5], Schechtman, and Schmuckenschläger [Reference Schechtman and Schmuckenschläger32]. Another ingredient in our estimates is a Bonnnesen-type inequality by Bokowski and Heil [Reference Bokowski and Heil7] on quermassintegrals of K. Finally, we use upper bounds on
$\theta (B_2^n)$
for specific small n which arise from known lattice coverings.
Theorem 1.1 is proved in Sections 2–4. Of possibly independent interest are estimates of the mean width of the regular simplex in dimensions
$5\le n\le 8$
given in Section 3.
In Section 5, we show how one can improve (1.5) for each fixed n by optimizing choices of certain parameters in the original proof of Rogers [Reference Rogers28] (the original proof provides a succinct bound valid for all n). Consequently, (1.1) and (1.4) can be somewhat improved for n larger than those covered by Theorem 1.1.
To finalize, in Tables 1 and 2 we provide the best known upper bounds on
$H_n$
and
$H_n^s$
for
$3\le n\le 14$
.
Best known upper bounds on
$H_n$
for
$3\le n\le 14$
.
Best known upper bounds on
$H_n^s$
for
$3\le n\le 14$
.
2 Preliminaries
In what follows,
$B_2^{n}=\{x\in {\mathbb {R}}^{n} \;: \|x\|\leq 1\}$
, and
${\mathbb {S}}^{n-1}=\{x\in {\mathbb {R}}^{n} \;: \|x\|= 1\}.$
The n-dimensional volume is denoted by
$|\cdot |_n$
, and subscript is usually dropped if the value of n is clear from the context.
2.1 John’s position
Let
${\mathcal {J}}_n$
be the family of convex bodies
$K\in {\mathcal {K}}_n$
which are in John’s position, i.e.,
$B_2^n$
is the maximal volume ellipsoid of K (see, e.g., [Reference Brazitikos, Giannopoulos, Valettas and Vritsiou9, Section 1.5.1]). Similarly, define
${\mathcal {J}}_n^s$
to be the family of centrally symmetric convex bodies
$K\in {\mathcal {K}}_n^s$
which are in John’s position. For any
$K\in {\mathcal {K}}_n$
(or
$K\in {\mathcal {K}}_n^s$
) there exists an affine image of K in
${\mathcal {J}}_n$
(or in
${\mathcal {J}}_n^s$
).
John’s theorem (see e.g., [Reference Artstein-Avidan, Giannopoulos and Milman2, Theorem 2.1.3, Remark 2.1.17]) implies that
$$ \begin{align} K\subseteq nB_2^n {\quad\text{for any}\quad} K\in{\mathcal{J}}_n, {\quad\text{and}\quad} K\subseteq \sqrt{n} B_2^n {\quad\text{for any}\quad} K\in {\mathcal{J}}_n^s. \end{align} $$
2.2 Quermassintegrals
In order to bound
$H_n$
and
$H_{n}^{s}$
we will use the inequality (1.8), and so we start by discussing upper bounds on
$|K+B_2^n|$
, where K is in John’s position.
By Steiner’s formula, for any
$K\in {\mathcal {K}}_n$
$$ \begin{align} |K+tB_2^n|=\sum_{j=0}^n \binom nj W_j(K) t^j, \end{align} $$
where
$$\begin{align*}W_j(K)=V(\underset{n-j \text{ times}}{\underbrace{K,\dots,K}}, \underset{j \text{ times}}{\underbrace{B_2^n,\dots,B_2^n}}) \end{align*}$$
is the jth quermassintegral of K and
$V(\cdot )$
is the mixed volume, see, e.g., [Reference Artstein-Avidan, Giannopoulos and Milman2, Section 1.1.5], [Reference Brazitikos, Giannopoulos, Valettas and Vritsiou9, Section1.4.2]or [Reference Schneider33, Section4.2]. Additionally,
$W_0(K)=|K|$
,
$W_1(K)=\partial (K)/n$
, where
$\partial (K)$
is the surface area of K, and
$W_n(K)=|B_2^n|$
. Also note that if
$B_2^n\subseteq K$
, then due to the monotonicity of mixed volumes we have
$W_i(K)\ge W_j(K)$
for
$i\le j$
.
Next, we will discuss upper bounds on
$W_i(K)$
for
$K\in {\mathcal {J}}_n$
and
$K\in {\mathcal {J}}_{n}^s$
respectively.
Let
$T^n$
be a regular simplex in
${\mathbb {R}}^n$
of unit edge length, and let
$\Delta ^n$
be a dilation of
$T^n$
for which
$B_2^n$
is the inscribed ball, then
$\Delta ^{n}=\sqrt {2(n+1)n} \; T^{n}$
. Also, let
$C^n:=[-1,1]^n$
be a cube circumscribed about
$B^2_n$
. Ball [Reference Ball3] proved that among all convex bodies in
${\mathbb {R}}^n$
simplexes have maximal volume ratio (volume ratio measures how much of the volume of the whole body can be contained in the largest inscribed ellipsoid) while for the symmetric ones such a maximizer is the cube [Reference Ball4]. These results can be stated in our terms as follows:
$$ \begin{align} W_0(K)\le \begin{cases} W_0(\Delta^n), &\text{if } K\in{\mathcal{J}}_n,\\ W_0(C^n), &\text{if } K\in{\mathcal{J}}_n^s. \end{cases} \end{align} $$
For
$K\in {\mathcal {K}}_n$
and a direction
$u\in {\mathbb {S}}^{n-1}$
, the support function is defined as
$ h_K(u)=\sup \{\langle x,y\rangle : y\in K\} $
. Let
$\sigma $
be the rotationally invariant probability measure on
${\mathbb {S}}^{n-1}$
, then the mean width of K is defined by
$$\begin{align*}w(K)=\int_{{\mathbb{S}}^{n-1}} h_K(u)\,d\sigma(u). \end{align*}$$
WeFootnote
1
have
$W_{n-1}(K)=|B_2^n|w(K)$
, which is a partial case of Kubota’s formula [Reference Artstein-Avidan, Giannopoulos and Milman2, eq (1.1.1)]. Barthe [Reference Barthe5, Theorem 3] provedFootnote
2
that among the bodies from
${\mathcal {J}}_n$
, the mean width is maximized for
$\Delta ^n$
, while Schechtman and Schmuckenschläger [Reference Schechtman and Schmuckenschläger32] (the proof is also included in [Reference Barthe5, Theorem 2]) remarked that
$C^n$
maximizes the mean width among the bodies from
${\mathcal {J}}_n^s$
, i.e.,
$$ \begin{align} W_{n-1}(K)\le \begin{cases} W_{n-1}(\Delta^n), &\text{if } K\in{\mathcal{J}}_n,\\ W_{n-1}(C^n), &\text{if } K\in{\mathcal{J}}_n^s. \end{cases} \end{align} $$
We estimate
$w(\Delta ^n)$
for the required values of n in the next section. It is known [Reference Finch15], [Reference Goodman and O’Rourke17, Section 13.2.3] (or can be obtained by a straightforward computation) that
$w(C^n)=2\frac {|B_2^{n-1}|_{n-1}}{|B_{2}^{n}|_n}$
for
$n\geq 2$
, and so
$W_{n-1}(C^{n})=2|B_{2}^{n-1}|$
.
Remark 2.1 The results [Reference Ball3, Theorem 1, Theorem 1′]imply that
$W_1(K)\le W_1(\Delta ^n)$
for any
$K\in {\mathcal {J}}_n$
(recall that
$W_1(K)=\partial (K)/n$
, where
$\partial (K)$
is the surface area of K). This would not lead to any improvements in our context as
$W_1(\Delta ^n)=W_0(\Delta ^n)$
and we get the same upper bound on
$W_1(K)$
as from
$W_1(K)\le W_0(K)$
. A similar remark regarding the inequality
$W_1(K)\leq W_1(C^{n})$
also holds for
$K \in {\mathcal {J}}^s_n$
, and follows from [Reference Ball3, Theorem 2], [Reference Ball4, Theorem 3].
Remark 2.2 It is natural to conjecture that
$W_j(K)\le W_j(\Delta ^n)$
(and that
$W_j(K)\le W_j(C^n)$
) for any
$K\in {\mathcal {J}}_n$
(respectively,
$K\in {\mathcal {J}}_n^s$
) and all
$0\le j\le n$
. This is an obvious equality for
$j=n$
and is valid for
$j\in \{0,1,n-1\}$
as described above ((2.3), Remark 2.1, (2.4)).
Finally, we need a Bonnesen-style inequality by Bokowski and Heil [Reference Bokowski and Heil7]. If
$K\in {\mathcal {K}}_n$
satisfies
$K\subseteq R B_2^n$
, then for all
$0\le i<j<k\le n$
$$ \begin{align} W_j(K) \le \frac{(k-j)(i+1)R^iW_i(K)+(j-i)(k+1)R^kW_k(K)}{(k-i)(j+1)R^j}=:B_{R,i,j,k}(W_i(K),W_k(K)). \end{align} $$
Recall that in our settings we can choose R according to (2.1).
2.3 Density of coverings by balls
One of the approaches to construct specific efficient coverings of the space by balls is to use lattices, see [Reference Conway and Sloane11, Chapter 2]. Considering the
$A_n^{*}$
lattice yields the following estimate (valid for all n):
$$ \begin{align} \theta(B_2^n)\le |B_2^n|\sqrt{n+1}\left(\frac{n(n+2)}{12(n+1)}\right)^{n/2}. \end{align} $$
It turns out that the above is optimal (smallest possible lattice covering density) for
$2\le n\le 5$
and provides the best known upper bound on
$\theta (B_2^n)$
in most dimensions
$10\le n\le 21$
. Better lattices were found by Schürmann and Vallentin [Reference Schürmann and Vallentin34] for
$6\le n\le 8$
, which is important for our applications. To the best of our knowledge, there have been no improvements after the work [Reference Schürmann and Vallentin34], where an interested reader can find a brief survey of the topic. We list the corresponding lattice covering densities of
${\mathbb {R}}^n$
by balls in Table 3.
Least known lattice covering densities, as in [Reference Schürmann and Vallentin34].
3 Estimates on mean width of regular simplex
The values of
$2w(T^n)$
for
$2\le n\le 6$
expressed as certain integrals and numerically evaluated with high precision can be found in [Reference Finch15]. We remark that in [Reference Finch15] expected values of widths of simplexes are considered, while the mean width as we defined here (and as commonly defined in the literature on geometry, see [Reference Artstein-Avidan, Giannopoulos and Milman2, Reference Brazitikos, Giannopoulos, Valettas and Vritsiou9]) is the expected value of the support function, this discrepancy results in the mean width of
$T^{n}$
from [Reference Finch15] (and [Reference Sun35]) being equal to
$2w(T^{n})$
. We need upper estimates of
$w(T^n)$
for
$n=7,8$
, which we were unable to find in the literature. We present a simple computational technique to obtain upper and lower estimates on
$w(T^n)$
which will suffice for our purposes.
The starting point is the following representation that follows the work [Reference Finch15] by Finch and references therein, in particular, Sun [Reference Sun35].
Let
$ F(x)=\frac {1}{\sqrt {2\pi }}\int _{-\infty }^{x}e^{-t^2/2}\; dt$
be the cumulative distribution function of the standard normal distribution, then
$F(x)=\frac {1}{2}+\frac {1}{2}\textrm {erf}(\frac {x}{\sqrt {2}})$
, where
$\textrm { erf}(z)=\frac 2{\sqrt {\pi }}\int _0^ze^{-t^2}\,dt$
is the error function.
For
$n\geq 0$
let
$$ \begin{align*} g_{n+1}(x)&:=1-F(x)^{n+1}-(1-F(x))^{n+1}= 1-\left(\frac{1+\textrm{erf}(\tfrac{x}{\sqrt{2}})}2\right)^{n+1}-\left(\frac{1-\textrm{erf}(\tfrac{x}{\sqrt{2}})}2\right)^{n+1}. \end{align*} $$
The following formula for mean width follows from [Reference Sun35] and was derived in [Reference Finch15]:
$$ \begin{align} w(T^n)= & \frac{\Gamma(\tfrac n2)}{2\Gamma(\tfrac {n+1}2)}\int_0^\infty g_{n+1}(x)\,dx. \end{align} $$
Next, we estimate the integral in the formula.
Proposition 3.1 For any
$a>2$
and any positive integers n, N, the following estimates hold:
$$ \begin{align} \frac aN\sum_{k=1}^N g_{n+1}\left(\tfrac{ak}N\right) \le \int_0^\infty g_{n+1}(x)\,dx \le \frac aN \sum_{k=0}^{N-1} g_{n+1}\left(\tfrac{ak}N\right) +\frac{n+1}{\sqrt{2\pi}} \exp\left(-a\right). \end{align} $$
Proof We directly estimate the “tail” of the integral and use simple endpoint Riemann sums for the “main” part of the integral in (3.1).
Recalling that
$\frac {d}{dx}F(x)=\frac {1}{\sqrt {2\pi }}e^{-x^2/2}$
it is straightforward to verify that
$g_{n+1}$
is a positive strictly decreasing function on
$[0,\infty )$
. Hence, considering the upper and the lower Riemann sums for
$\int _0^a g_{n+1}(x)\,dx$
and the uniform partition of
$[0,a]$
into N subintervals, we obtain
$$ \begin{align} \frac aN\sum_{k=1}^N g_{n+1}\left(\tfrac{a\cdot k}N\right) \le \int_0^a g_{n+1}(x)\,dx \le \frac aN \sum_{k=0}^{N-1} g_{n+1}\left(\tfrac{a\cdot k}N\right). \end{align} $$
For any
$x>a\geq 2$
$$ \begin{align*} g_{n+1}(x) &\le 1-F(x)^{n+1} = 1-\left(1-F(-x)\right)^{n+1}=1-\left(1-\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^2/2}\,dt\right)^{n+1} \\ &\le \frac{n+1}{\sqrt{2\pi}}\int_{x}^\infty e^{-t^2/2}\,dt \le \frac{n+1}{\sqrt{2\pi}}\int_{x}^\infty e^{-t}\,dt =\frac{n+1}{\sqrt{2\pi}}\exp\left(-x \right), \end{align*} $$
so
$$ \begin{align*} \int_a^\infty g_{n+1}(x)\,dx \le \frac{n+1}{\sqrt{2\pi}} \exp\left(-a\right). \end{align*} $$
Taking this inequality and the evident
$\int _a^\infty g_{n+1}(x)\,dx\ge 0$
into account, we deduce (3.2) from (3.1) and (3.3).
Since the error function can be computed numerically with any given precision, employing a simple SageMath [31] computation [Reference Arman, Bondarenko and Prymak1], we obtain the following corollary.
Corollary 3.2 The following inequalities hold:
$$ \begin{align*} 0.4208 \le & w(T^5) \le 0.4215, \\ 0.4067 \le & w(T^6) \le 0.407, \\ 0.39425 \le & w(T^7) \le 0.39427, \\ 0.383 \le & w(T^8) \le 0.38301. \end{align*} $$
These estimates for
$n=5,6$
are consistent with the values obtained in [Reference Finch15]; we include them here for completeness. While our computational method allows to obtain a tighter gap in the estimates, we only derived what was necessary for our application of estimating
$H_n$
from above ensuring that no further improvement is possible (even if the value of the lower bound on
$w(T^n)$
is used, the upper bound on the integer value
$H_n$
does not improve). The computations take less than two hours on a modern personal computer.
4 Proof of Theorem 1.1
Let us begin with the general (not necessarily symmetric) case. Since
$C(K,\textrm {int}(K))$
is invariant under affine transforms, we can assume that
$K\in {\mathcal {J}}^n$
. Then using
$B_2^n\subseteq K$
, compactness and (1.6), we have
$$ \begin{align*} C(K,\textrm{int}(K))&\le C(K,\textrm{int}(B_2^n))\le \lim_{r\to1^-} C(K,rB_2^n) \\ &\le \lim_{r\to1^-} \frac{|K-rB_2^n|}{|rB_2^n|}\theta(B_2^n) = \frac{|K+B_2^n|}{|B_2^n|}\theta(B_2^n). \end{align*} $$
Hence, applying (2.2),
$$ \begin{align} C(K,\textrm{int}(K))\le \frac{\theta(B_2^n)}{|B_2^n|} \sum_{j=0}^n \binom nj W_j(K). \end{align} $$
The upper bounds for
$\theta (B_2^n)$
come from Table 3, so it remains to estimate
$W_i(K)$
. As
$\Delta ^n=\sqrt {2n(n+1)}T^n$
, by (2.3),
$$ \begin{align} W_i(K)\le W_0(K)\le W_0(\Delta^n)=(2n(n+1))^{n/2}|T^n|=\frac{n^{n/2}(n+1)^{(n+1)/2}}{n!} \end{align} $$
for any
$0\le i\le n$
. Using (2.4),
$$ \begin{align} W_{n-1}(K)\le W_{n-1}(\Delta^n)=|B_2^n|w(\Delta^n)=|B_2^n|\sqrt{2n(n+1)}w(T^n). \end{align} $$
Trivially,
$W_n(K)=|B_2^n|$
.
Next we combine the above, and use (2.5) (valid with
$R=n$
due to (2.1)) for appropriate parameters. The calculations were performed in the script [Reference Arman, Bondarenko and Prymak1].
Bound on
$H_5$
. We apply (4.1) and estimate the quermassintegrals as follows. For
$0\le i\le 2$
, use (4.2); (4.3) and (2.5) give
$W_4(K)\le W_4(\Delta ^5)$
and
$W_3(K)\le B_{5,2,3,4}(W_0(\Delta ^5),W_4(\Delta ^5))$
; recall that
$W_5(K)=|B_2^5|$
. Combining the above, using Corollary 3.2 and calculating the actual value of the bound yields
$H_5\le 933$
.
Bound on
$H_6$
. The arguments are similar to the previous case. The required application of (2.5) is
$W_4(K)\le B_{6,3,4,5}(W_0(\Delta ^6),W_5(\Delta ^6))$
. The result is
$H_6\le 6137$
.
Bounds on
$H_n$
for
$n=7,8$
. We proceed similarly with the only difference that
$W_j(K)\le B_{n,n-4,j,n-1}(W_0(\Delta ^n),W_{n-1}(\Delta ^n))$
is used for
$j\in \{n-3,n-2\}$
yielding
$H_7\le 41377$
and
$H_8\le 284096$
.
When the body is centrally symmetric, we follow a similar route, with the following differences. In place of (4.2), we have
$$ \begin{align} W_i(K)\le W_0(K)\le W_0(C^n)=2^n \end{align} $$
for any
$0\le i\le n$
. Using (2.4) we get
$$ \begin{align} W_{n-1}(K)\le W_{n-1}(C^{n})=2|B_2^{n-1}|. \end{align} $$
Finally, by (2.1), the inequality (2.5) can be used with
$R=\sqrt {n}$
.
Bound on
$H_4^s$
. Apply (4.1) and bound the quermassintegrals as follows. For
$i=0,1$
use (4.4); use (4.5) to get
$W_{3}(K)\leq 2|B_{2}^{3}|$
; we have
$W_4(K)=W_4(C^{4})=|B_2^4|$
; by (2.5)
$W_2(K)\le B_{2,1,2,3}(W_1(C^4),W_3(C^4))$
. Combining the above and calculating the value of the bound yields
$H_4^s\le 72$
.
Bounds on
$H_n^s$
for
$n=5,6$
. We use similar arguments: (4.4) for
$i=0,1,2$
;
$W_n(K)=W_n(C^n)=|B_{2}^{n}|$
; (4.5) for
$i=n-1$
; and
$W_j(K)\le B_{\sqrt {n},2,j,n-1}(W_0(C^n),W_{n-1}(C^n))$
for
$3\le j\le n-2$
, which imply
$H_5^s\le 305$
and
$H_6^s\le 1292$
.
Remark 4.1 In the above computations, we used the exact value for the density arising from
$A_n^*$
lattice from the estimate (2.6) for
$n=4,5$
. For
$n=6,7,8$
we used the values of covering densities from [Reference Schürmann and Vallentin34] presented in Table 3 increased by
$5\cdot 10^{-6}$
as they were given to 6 decimal places, while we require an upper bound. Even if such an increase is not performed, the resulting integer valued upper bounds in Theorem 1.1 would not change, so the accuracy given in [Reference Schürmann and Vallentin34] is more than sufficient for our needs.
5 Estimates via optimized Rogers bound
Proposition 5.1 Suppose
$n\ge 3$
. If
$$ \begin{align} r_n=\min_{x\in(0,1/n)} f_n(x), {\quad\text{where}\quad} f_n(x)=(1+x)^n(1-n\ln(x)), \end{align} $$
then
$$ \begin{align} \theta(K)\le r_n {\quad\text{for any}\quad} K\in {\mathcal{K}}_n, \quad H_n\le \binom{2n}{n} r_n, {\quad\text{and}\quad} H_n^s\le 2^n r_n. \end{align} $$
Proof The first inequality in (5.2) is established in [Reference Rogers28, p. 5], p. 5. (The bound (1.5) was obtained in [Reference Rogers28] by taking
$x=\frac 1{n\ln n}$
in (5.1).) The other two inequalities in (5.2) follow from (1.6) in the same way as (1.1) and (1.4) do.
For our purposes, it suffices to use the straightforward
$r_n\le \min \{ f_n(\frac {j}{Nn}):1\le j\le N-1\}$
with
$N=1000$
. The resulting bounds on
$r_n$
for
$3\le n\le 14$
are given in Table 4 (the computed values of
$\min \{ f_n(\frac {j}{Nn}):1\le j\le N-1\}$
are rounded up in the sixth digit, so they represent actual upper bounds), and the estimates on
$H_n$
and
$H_n^s$
can be found in Tables 1 and 2. For the computations, see [Reference Arman, Bondarenko and Prymak1]. We remark that
$\max \{\theta (K), K\in {\mathcal {K}}_2\}=\frac 32$
was established by Fáry [Reference Fáry14].
Upper bounds on
$\max \{\theta (K),\ K\in {\mathcal {K}}_n\}$
, for
$3\le n\le 14$
.
Acknowledegments
The authors are grateful to the referees for the numerous comments that improved the paper.
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