On the volume of unit balls of finite-dimensional Lorentz spaces

TL;DR

This paper investigates the volume of unit balls in finite-dimensional Lorentz spaces, providing formulas, asymptotic behavior, and applications to entropy numbers, advancing understanding of geometric properties of these function spaces.

Contribution

It introduces explicit formulas and asymptotic estimates for the volume of unit balls in Lorentz spaces, including weak Lebesgue spaces, and applies these to entropy number decay analysis.

Findings

Derived iterative formulas for volume in weak Lebesgue spaces

Established asymptotic behavior of volume as dimension grows

Characterized entropy number decay for specific embeddings

Abstract

We study the volume of unit balls $B^n_{p,q}$ of finite-dimensional Lorentz sequence spaces $\ell_{p,q}^n.$ We give an iterative formula for ${\rm vol}(B^n_{p,q})$ for the weak Lebesgue spaces with $q=\infty$ and explicit formulas for $q=1$ and $q=\infty.$ We derive asymptotic results for the $n$-th root of ${\rm vol}(B^n_{p,q})$ and show that $[{\rm vol}(B^n_{p,q})]^{1/n}\approx n^{-1/p}$ for all $0<p<\infty$ and $0<q\le\infty.$ We study further the ratio between the volume of unit balls of weak Lebesgue spaces and the volume of unit balls of classical Lebesgue spaces. We conclude with an application of the volume estimates and characterize the decay of the entropy numbers of the embedding of the weak Lebesgue space $\ell_{1,\infty}^n$ into $\ell_1^n.$

Tables

Table 1. Table 1: vol ​ ( B p , ∞ n ) vol subscript superscript 𝐵 𝑛 𝑝 \text{vol}(B^{n}_{p,\infty}) for dimensions up to 15

$n$	$p=1/2$		$p=1$		$p=2$		$p=100$
$1$	$2.000$	$\cdot {10}^0$	$2.000$	$\cdot {10}^0$	$2.000$	$\cdot {10}^0$	$2.000$	$\cdot {10}^0$
$2$	$1.333$	$\cdot {10}^0$	$2.000$	$\cdot {10}^0$	$2.343$	$\cdot {10}^0$	$2.660$	$\cdot {10}^0$
$3$	$4.444$	$\cdot {10}^{-1}$	$1.333$	$\cdot {10}^0$	$2.051$	$\cdot {10}^0$	$2.891$	$\cdot {10}^0$
$4$	$8.889$	$\cdot {10}^{-2}$	$6.667$	$\cdot {10}^{-1}$	$1.473$	$\cdot {10}^0$	$2.761$	$\cdot {10}^0$
$5$	$1.185$	$\cdot {10}^{-2}$	$2.667$	$\cdot {10}^{-1}$	$9.119$	$\cdot {10}^{-1}$	$2.404$	$\cdot {10}^0$
$6$	$1.129$	$\cdot {10}^{-3}$	$8.889$	$\cdot {10}^{-2}$	$5.010$	$\cdot {10}^{-1}$	$1.949$	$\cdot {10}^0$
$7$	$8.062$	$\cdot {10}^{-5}$	$2.540$	$\cdot {10}^{-2}$	$2.494$	$\cdot {10}^{-1}$	$1.492$	$\cdot {10}^0$
$8$	$4.479$	$\cdot {10}^{-6}$	$6.349$	$\cdot {10}^{-3}$	$1.141$	$\cdot {10}^{-1}$	$1.089$	$\cdot {10}^0$
$9$	$1.991$	$\cdot {10}^{-7}$	$1.411$	$\cdot {10}^{-3}$	$4.851$	$\cdot {10}^{-2}$	$7.632$	$\cdot {10}^{-1}$
$10$	$7.239$	$\cdot {10}^{-9}$	$2.822$	$\cdot {10}^{-4}$	$1.932$	$\cdot {10}^{-2}$	$5.164$	$\cdot {10}^{-1}$
$11$	$2.194$	$\cdot {10}^{-10}$	$5.131$	$\cdot {10}^{-5}$	$7.260$	$\cdot {10}^{-3}$	$3.387$	$\cdot {10}^{-1}$
$12$	$5.625$	$\cdot {10}^{-12}$	$8.551$	$\cdot {10}^{-6}$	$2.588$	$\cdot {10}^{-3}$	$2.161$	$\cdot {10}^{-1}$
$13$	$1.236$	$\cdot {10}^{-13}$	$1.316$	$\cdot {10}^{-6}$	$8.789$	$\cdot {10}^{-4}$	$1.345$	$\cdot {10}^{-1}$
$14$	$2.355$	$\cdot {10}^{-15}$	$1.879$	$\cdot {10}^{-7}$	$2.856$	$\cdot {10}^{-4}$	$8.183$	$\cdot {10}^{-2}$
$15$	$3.924$	$\cdot {10}^{-17}$	$2.506$	$\cdot {10}^{-8}$	$8.904$	$\cdot {10}^{-5}$	$4.877$	$\cdot {10}^{-2}$

Equations

$$\|x\|_{p}=\begin{cases}\displaystyle\Bigl{(}\sum_{j=1}^{n}|x_{j}|^{p}\Bigr{)}^{1/p},&\quad\text{for}\ 0<p<\infty,\\ \displaystyle\max_{j=1,\dots,n}|x_{j}|,&\quad\text{for}\ p=\infty\end{cases}$$

$$B_p^n=\{x\in {\mathbb{R}}^n:\Vert x{\Vert }_p\le 1\}.$$

$$\Vert x{\Vert }_{p,q}=\Vert k^{\frac{1}{p}-\frac{1}{q}}{x}_k^{\ast }{\Vert }_q,$$

$$B_{p,q}^n=\{x\in {\mathbb{R}}^n:\Vert x{\Vert }_{p,q}\le 1\}.$$

$$\operatornamewithlimits{vol}(B_{p}^{n})=2^{n}\cdot\frac{\Gamma\bigl{(}1+\frac{1}{p}\bigr{)}^{n}}{\Gamma\bigl{(}1+\frac{n}{p}\bigr{)}}.$$

$$\mathrm{vol}(B_{p,\infty }^{n,+})=\sum _{j=1}^n(-1{)}^{j-1}(\genfrac{}{}{0ex}{}{n}{j})n^{-j/p}\mathrm{vol}(B_{p,\infty }^{n-j,+})$$

$$\mathrm{vol}(B_{p,1}^n)=2^n\prod _{k=1}^n\frac{1}{{\varkappa }_p(k)},\text{where}{\varkappa }_p(k)=\sum _{j=1}^k{j}^{1/p-1}.$$

$$e_k(id:{\ell }_{1,\infty }^n\to {\ell }_1^n)\approx \{\begin{array}{l}\log (1+n/k),1\le k\le n,\\ 2^{-\frac{k-1}{n}},k\ge n\end{array}$$

$$\mathrm{vol}(B_{p,\infty }^{1,+})=\mathrm{vol}(B_{p,\infty }^{0,+})=1$$

$$\mathrm{vol}(B_{p,\infty }^{n,+})=\sum _{j=1}^n(-1{)}^{j-1}(\genfrac{}{}{0ex}{}{n}{j})n^{-j/p}\mathrm{vol}(B_{p,\infty }^{n-j,+}).$$

$$B_{p,\infty }^{n,+}=\bigcup _{j=1}^n{A}_j.$$

$$A_K=\bigcap _{k\in K}{A}_k=\{x\in B_{p,\infty }^{n,+}:x_k\le n^{-1/p}\text{for all}k\in K\}.$$

$$\operatornamewithlimits{vol}(A_{K})=\Bigl{(}\prod_{k\in K}n^{-1/p}\Bigr{)}\cdot\operatornamewithlimits{vol}(B^{n-|K|,+}_{p,\infty})=n^{-|K|/p}\cdot\operatornamewithlimits{vol}(B^{n-|K|,+}_{p,\infty}).$$

$$\mathrm{vol}(B_{p,\infty }^{n,+})$$

$$=\sum _{\varnothing \ne K\subset \{1,\dots ,n\}}(-1{)}^{|K|-1}{n}^{-|K|/p}\mathrm{vol}(B_{p,\infty }^{n-|K|,+})$$

$$=\sum _{j=1}^n(-1{)}^{j-1}(\genfrac{}{}{0ex}{}{n}{j})n^{-j/p}\mathrm{vol}(B_{p,\infty }^{n-j,+}).$$

$${\mathbf{K}}_n$$

$${\mathbf{M}}_n$$

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\left(\genfrac{}{}{0ex}{}{n}{k_1,\dots ,k_{\ell (k)}}\right)=\frac{n!}{k_1!\dots k_{\ell (k)}!}$$

$$\mathrm{vol}(B_{p,\infty }^{n,+})$$

$$=n!\sum _{m\in {\mathbf{M}}_n}(-1{)}^{n+\ell (m)}\prod _{l=0}^{\ell (m)-1}\frac{(n-m_l{)}^{-(m_{l+1}-m_l)/p}}{(m_{l+1}-m_l)!}.$$

$$\mathrm{vol}(B_{p,\infty }^{n,+})$$

$$\displaystyle=\sum_{j=1}^{n}(-1)^{j-1}{n\choose j}n^{-j/p}\sum_{k\in{\bf K}_{n-j}}(-1)^{n-j+\ell(k)}{n-j\choose k}\prod_{l=1}^{\ell(k)}\Bigl{(}n-j-\sum_{i=1}^{l-1}k_{i}\Bigr{)}^{-k_{l}/p}$$

$$\displaystyle=\sum_{j=1}^{n}\sum_{k\in{\bf K}_{n-j}}(-1)^{n+\ell(k)-1}{n\choose j}{n-j\choose k}n^{-j/p}\prod_{l=1}^{\ell(k)}\Bigl{(}n-j-\sum_{i=1}^{l-1}k_{i}\Bigr{)}^{-k_{l}/p}$$

$$\displaystyle=\sum_{\nu\in{\bf K}_{n}}(-1)^{n+\ell(\nu)}{n\choose\nu}n^{-\nu_{1}/p}\prod_{l=1}^{\ell(\nu)-1}\Bigl{(}n-\sum_{i=1}^{l}\nu_{i}\Bigr{)}^{-\nu_{l+1}/p}$$

$$\displaystyle=\sum_{\nu\in{\bf K}_{n}}(-1)^{n+\ell(\nu)}{n\choose\nu}\prod_{l=1}^{\ell(\nu)}\Bigl{(}n-\sum_{i=1}^{l-1}\nu_{i}\Bigr{)}^{-\nu_{l}/p},$$

$$V^{(m)}(n,a)={\int }_0^{a_n}{\int }_{x_n}^{a_{n-1}}\cdots {\int }_{x_2}^{a_1}{x}_1^{m}d{x}_1\dots d{x}_{n-1}d{x}_n.$$

$$0\le x_n\le a_n,x_n\le x_{n-1}\le a_{n-1},\dots ,x_2\le x_1\le a_1,$$

$$0\le x_n\le x_{n-1}\le \cdots \le x_2\le x_1\text{and}{x}_j\le a_j\text{for all}j=1,\dots ,n.$$

$$\mathrm{vol}(B_{p,\infty }^{n,+})=n!\cdot V^{(0)}(n,a^{(p)}).$$

$$V^{(m)}(n,a)=\sum _{i=1}^n(-1{)}^{i+1}\frac{a_i^{m+i}m!}{(m+i)!}{V}^{(0)}(n-i,(a_{i+1},\dots ,a_n)).$$

$$V^{(m)}(n,a)$$

$$=\frac{1}{m+1}{\int }_0^{a_n}{\int }_{x_n}^{a_{n-1}}\cdots {\int }_{x_3}^{a_2}(a_1^{m+1}-x_2^{m+1})d{x}_2\dots d{x}_{n-1}d{x}_n$$

$$=\frac{a_1^{m+1}}{m+1}{V}^{(0)}(n-1,(a_2,\dots ,a_n))-\frac{1}{m+1}{V}^{(m+1)}(n-1,(a_2,\dots ,a_n)).$$

$$V^{(m)}(n,a)$$

Full text

Keywords: Lorentz sequence spaces, weak Lebesgue spaces, entropy numbers, volume estimates, (non-)convex bodies

1 Introduction and main results

It was observed already in the first half of the last century (cf. the interpolation theorem of Marcinkiewicz [34]) that the scale of Lebesgue spaces $L_{p}(\Omega)$, defined on a subset $\Omega\subset\mathbb{R}^{n}$, is not sufficient to describe the fine properties of functions and operators. After the pioneering work of Lorentz [31, 32], Hunt defined in [23, 24] a more general scale of function spaces $L_{p,q}(\Omega)$, the so-called Lorentz spaces. This scale includes Lebesgue spaces as a special case (for $p=q$) and Lorentz spaces have found applications in many areas of mathematics, including harmonic analysis (cf. [19, 20]) and the analysis of PDE’s (cf. [28, 35]).

If $\Omega$ is an atomic measure space (with all atoms of the same measure), one arrives naturally at the definition of Lorentz spaces on (finite or infinite) sequences. If $n$ is a positive integer and $0<p\leq\infty$, then the Lebesgue $n$-dimensional space $\ell_{p}^{n}$ is $\mathbb{R}^{n}$ equipped with the (quasi-)norm

[TABLE]

for every $x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$. We denote by $B_{p}^{n}$ its unit ball

[TABLE]

If $0<p,q\leq\infty$, then the Lorentz space $\ell_{p,q}^{n}$ stands for $\mathbb{R}^{n}$ equipped with the (quasi-)norm

[TABLE]

where $x^{*}=(x_{1}^{*},\dots,x_{n}^{*})$ is the non-increasing rearrangement of $(|x_{1}|,\dots,|x_{n}|)$. If $p=q$, then $\ell_{p,p}^{n}=\ell_{p}^{n}$ are again the Lebesgue sequence spaces. If $q=\infty$, then the space $\ell_{p,\infty}^{n}$ is usually referred to as a weak Lebesgue space.

Similarly to (1), we denote by $B_{p,q}^{n}$ the unit ball of $\ell_{p,q}^{n}$, i.e. the set

[TABLE]

Furthermore, $B^{n,+}_{p}$ (or $B_{p,q}^{n,+}$) will be the set of vectors from $B_{p}^{n}$ (or $B_{p,q}^{n}$) with all coordinates non-negative.

Lorentz spaces of (finite or infinite) sequences have been used extensively in different areas of mathematics. They form a basis for many operator ideals of Pietsch, cf. [37, 38, 42], they play an important role in the interpolation theory, cf. [4, 5, 22, 30], and their weighted counterparts are the main building blocks of approximation function spaces, cf. [40, 43]. Weak Lebesgue sequence spaces (i.e. Lorentz spaces with $q=\infty$) were used by Cohen, Dahmen, Daubechies, and DeVore [8] to characterize functions of bounded variation. Lorentz spaces further appear in approximation theory [9, 10, 11] and signal processing [7, 13, 17].

The volume of unit balls of classical Lebesgue sequence spaces $B_{p}^{n}$ is known since the times of Dirichlet [12], who showed for $0<p\leq\infty$ that

[TABLE]

Here, $\operatornamewithlimits{vol}(A)$ stands for the Lebesgue measure of a (measurable) set $A\subset\mathbb{R}^{n}$ and $\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}dt$ is the Gamma function for $x>0$. Since then, (4) and its consequences play an important role in many results about finite-dimensional Lebesgue spaces, cf. [39]. Although many properties of Lorentz sequence spaces were studied in detail earlier (cf. [2, 3, 4, 5, 25, 29, 33]), there seems to be only very little known about the volume of their unit balls. The aim of this work is to fill to some extent this gap.

We present two ways, which lead to recursive formulas for $\operatornamewithlimits{vol}(B_{p,\infty}^{n})$ if $0<p<\infty$. The first one (cf. Theorem 1)

[TABLE]

is quite well suited for calculating $\operatornamewithlimits{vol}(B_{p,\infty}^{n})$ for moderate values of $n$ and we present also some numerical results on the behavior of this quantity for different values of $p$. Although an explicit formula for $\operatornamewithlimits{vol}(B_{p,\infty}^{n})$ can be derived from (5), cf. Theorem 2, due to its combinatorial nature it seems to be only of a limited practical use. In Section 2.1.2 we derive the same formula with the help of iterated multivariate integrals, very much in the spirit of the original proof of Dirichlet.

Surprisingly, a simple explicit formula can be given for $\operatornamewithlimits{vol}(B_{p,1}^{n})$ for the full range of $0<p\leq\infty$. Indeed, we show in Theorem 5 that

[TABLE]

If $p=1$, then $\varkappa_{1}(k)=k$ and this formula reduces immediately to the well-known relation $\operatornamewithlimits{vol}(B_{1}^{n})=2^{n}/n!.$

Using Stirling’s formula, (4) implies that $[\operatornamewithlimits{vol}(B_{p}^{n})]^{1/n}\approx n^{-1/p}$ for all $0<p<\infty$ with the constants of equivalence independent of $n$. Using the technique of entropy numbers, we show in Theorem 6 that essentially the same is true for the whole scale of Lorentz spaces $\ell_{p,q}^{n}$ (with a remarkable exception for $p=\infty$, cf. Theorem 7).

It is a very well known fact (cf. Theorem 8) that $B_{p}^{n}\subset B_{p,\infty}^{n}$ for all $0<p\leq\infty$ and it is a common folklore to consider the unit balls of weak Lebesgue spaces (i.e. Lorentz spaces with $q=\infty$) as the “slightly larger” counterparts of the unit balls of Lebesgue spaces with the same summability parameter $p$. This intuition seems to be further confirmed by Theorem 6, which shows that the quantities $[\operatornamewithlimits{vol}(B^{n}_{p})]^{1/n}$ and $[\operatornamewithlimits{vol}(B^{n}_{p,\infty})]^{1/n}$ are equivalent to each other with constants independent on $n$. On the other hand, we show in Theorem 9 that $\operatornamewithlimits{vol}(B^{n}_{p,\infty})/\operatornamewithlimits{vol}(B^{n}_{p})$ grows exponentially in $n$ at least for $p\leq 2$. We conjecture (but it remains an open problem) that the same is true for all $p<\infty.$

We conclude our work by considering the entropy numbers of the embeddings between Lorentz spaces of finite dimension, which complements the seminal work of Edmunds and Netrusov [14]. We characterize in Theorem 11 the decay of the entropy numbers $e_{k}(id:\ell_{1,\infty}^{n}\to\ell_{1}^{n})$, which turns out to exhibit a rather unusual behavior, namely

[TABLE]

with constants of equivalence independent of $k$ and $n$. We see that after a logarithmic decay for $1\leq k\leq n$, the exponential decay in $k$ takes over for $k\geq n.$

2 Recursive and explicit formulas

In this section, we present different formulas for the volume of unit balls of Lorentz spaces for two special cases, namely for the weak Lebesgue spaces with $q=\infty$ and for Lorentz spaces with $q=1.$ Surprisingly, different techniques have to be used in these two cases.

2.1 Weak Lebesgue spaces

We start by the study of weak Lebesgue spaces, i.e. the Lorentz spaces $\ell_{p,\infty}^{n}$. If $p=\infty$, then $\ell_{p,\infty}^{n}=\ell_{\infty}^{n}$. Therefore, we restrict ourselves to $0<p<\infty$ in this section.

2.1.1 Using the inclusion-exclusion principle

In this section, we assume the convention

[TABLE]

for every $0<p<\infty.$

Theorem 1**.**

Let $n\in\mathbb{N}$ and $0<p<\infty$. Then

[TABLE]

Proof.

For $1\leq k\leq n$, we denote $A_{k}=\{x\in B^{n,+}_{p,\infty}:x_{k}\leq n^{-1/p}\}$. If $x\in B^{n,+}_{p,\infty}$, then at least one of the coordinates of $x$ must be smaller than or equal to $n^{-1/p}$. Therefore

[TABLE]

For a non-empty index set $K\subset\{1,\dots,n\}$, we denote

[TABLE]

If we denote by $x_{K^{c}}$ the restriction of $x$ onto $K^{c}=\{1,\dots,n\}\setminus K$, then the $j^{\rm th}$ largest coordinate of $x_{K^{c}}$ can be at most $j^{-1/p}$, i.e. $x_{K^{c}}\in B^{n-|K|,+}_{p,\infty}$. Here, $|K|$ stands for the number of elements in $K$. We therefore obtain

[TABLE]

Finally, we insert (7) into the inclusion-exclusion principle and obtain

[TABLE]

∎

The relation (6) is already suitable for calculation of $\operatornamewithlimits{vol}(B^{n}_{p,\infty})$ for moderate values of $n$, cf. Table 1. Let us remark, that $\operatornamewithlimits{vol}(B^{n}_{1,\infty})$ is maximal for $n=4$ and $\operatornamewithlimits{vol}(B^{n}_{2,\infty})$ attains its maximum at $n=18.$

Next we exploit Theorem 1 to give a certain explicit result about the volume of unit balls of weak Lebesgue spaces. For this, we denote by ${\bf K}_{n}$ the set of integer vectors of finite length $k=(k_{1},\dots,k_{j})$ with positive coordinates $k_{1},\dots,k_{j}$, which sum up to $n$. We denote by $\ell(k)=j$ the length of $k\in{\bf K}_{n}.$ Similarly, we denote by ${\bf M}_{n}$ the set of all increasing sequences $m=(m_{0},\dots,m_{j})$ which grow from zero to $n$, i.e. with $0=m_{0}<m_{1}<\dots<m_{j}=n.$ The quantity $\ell(m)=j$ is again the length of $m\in{\bf M}_{n}$. Hence,

[TABLE]

For $k\in{\bf K}_{n}$, we also write

[TABLE]

The explicit formula for $\operatornamewithlimits{vol}(B_{p,\infty}^{n})=2^{n}\operatornamewithlimits{vol}(B_{p,\infty}^{n,+})$ is then presented in the following theorem.

Theorem 2**.**

Let $0<p<\infty$ and $n\in\mathbb{N}.$ Then

[TABLE]

Proof.

First, we prove the second identity in (8). Indeed, the mapping $k=(k_{1},\dots,k_{j})\to(0,k_{1},k_{1}+k_{2},\dots,\sum_{i=1}^{j}k_{i})$ maps one-to-one ${\bf K}_{n}$ onto ${\bf M}_{n}$, preserving also the length of the vectors.

Next, we proceed by induction to show the first identity of (8). For that sake, we denote ${\bf K}_{0}=\{0\}$ with $\ell(0)=0$. With this convention, (8) is true for both $n=0$ and $n=1$, where both the sides of (8) are equal to one. The rest follows from (6). Indeed, we obtain

[TABLE]

where we identified the pair $(j,k)$ with $1\leq j\leq n$ and $k=(k_{1},\dots,k_{\ell(k)})\in{\bf K}_{n-j}$ with $\nu=(j,k_{1},\dots,k_{\ell(k)})\in{\bf K}_{n}$. If $j=n$, then the pair $(n,0)$ is identified with $\nu=(n)$. In any case, $\ell(\nu)=\ell(k)+1.$ ∎

2.1.2 Integral approach

The result of Theorem 2 can be obtained also by an iterative evaluation of integrals, resembling the approach of the original work of Dirichlet [12]. To begin with, we define a scale of expressions which for some specific choice of parameters lead to a formula for $\operatornamewithlimits{vol}(B^{n}_{p,\infty})$.

Definition 3**.**

Let $m\in\mathbb{N}_{0}$ and $n\in\mathbb{N}$. Let $a\in\mathbb{R}^{n}$ be a decreasing positive vector, i.e. $a=(a_{1},\dots,a_{n})$ with $a_{1}>a_{2}>\dots>a_{n}>0.$ We denote

[TABLE]

The domain of integration in (9) is defined by the following set of conditions

[TABLE]

which can be reformulated also as

[TABLE]

Hence, the integration in (9) goes over the cone of non-negative non-increasing vectors in $\mathbb{R}^{n}$ intersected with the set $\{x\in\mathbb{R}^{n}:x^{*}_{j}\leq a_{j}\ \text{for}\ j=1,\dots,n\}.$ If we set $a^{(p)}=(a_{1}^{(p)},\dots,a_{n}^{(p)})$ with $a_{j}^{(p)}=j^{-1/p}$ for $0<p<\infty$ and $1\leq j\leq n$, this set coincides with $B^{n,+}_{p,\infty}$. Finally, considering all the possible reorderings of $x$, we get

[TABLE]

In what follows, we simplify the notation by assuming $V^{(0)}(0,\emptyset)=1.$ The integration in (9) leads to the following recursive formula for $V^{(m)}(n,a)$.

Lemma 4**.**

Let $m\in\mathbb{N}_{0}$, $n\in\mathbb{N}$ and $a\in\mathbb{R}^{n}$ with $a_{1}>a_{2}>\dots>a_{n}>0.$ Then

[TABLE]

Proof.

First, we obtain

[TABLE]

The proof of (11) now follows by induction over $n$. For $n=1$, $V^{(m)}(1,(a_{1}))=\frac{a_{1}^{m+1}}{m+1}$, which is in agreement with (11). To simplify the notation later on, we write $a_{[k,l]}=(a_{k},\dots,a_{l})$ for every $1\leq k\leq l\leq n$. We assume, that (11) holds for $n-1$ and calculate

[TABLE]

which finishes the proof of (11). ∎

Lemma 4 allows for a different proof of Theorem 2.

Alternative proof of Theorem 2.

By (10), we need to calculate $V^{(0)}(n,a)$ and then substitute $a=a^{(p)}.$ We show by induction that

[TABLE]

If $n=1$, then both sides of (12) are equal to $a_{1}.$ For general $n$, we obtain by (11)

[TABLE]

where we identified the pair $(i,m)$ with $1\leq i\leq n$ and $m=(m_{0},\dots,m_{\ell(m)})\in{\bf M}_{n-i}$ with $\mu=(\mu_{0},\dots,\mu_{\ell(\mu)})=(m_{0},\dots,m_{\ell(m)},n)\in{\bf M}_{n}$. Hence, $\ell(\mu)=\ell(m)+1.$ ∎

2.2 Lorentz spaces with $q=1$

We give an explicit formula for $\operatornamewithlimits{vol}(B_{p,1}^{n})$, which takes a surprisingly simple form. The approach is based on the polarization. Recall, that for $0<p\leq\infty$, the (quasi-)norm $\|x\|_{p,1}$ is defined as

[TABLE]

Theorem 5**.**

Let $0<p\leq\infty$. Then

[TABLE]

Proof.

Let $f:[0,\infty)\to\mathbb{R}$ be a smooth non-negative function with a sufficient decay at infinity (later on, we will just choose $f(t)=e^{-t}$). Then

[TABLE]

For the choice $f(t)=e^{-t}$, we get

[TABLE]

It remains to evaluate

[TABLE]

where

[TABLE]

We denote for $t\geq 0$, $0<p\leq\infty$, and $n\in\mathbb{N}$

[TABLE]

where {\mathcal{C}}_{t,+}^{n}=\Bigl{\{}x\in\mathbb{R}^{n}:x_{1}\geq x_{2}\geq\dots\geq x_{n}\geq t\Bigr{\}}, i.e. $I_{p}^{n}=2^{n}\cdot n!\cdot A(n,p,0).$ We observe that

[TABLE]

and

[TABLE]

Combining (15) and (16), we prove by induction

[TABLE]

and

[TABLE]

Finally, we combine (13) with (14) and (17) and obtain

[TABLE]

∎

Remark 1*.*

Let us point out that for $p=1$, we get $\varkappa_{1}(k)=k$ and we recover the very well known formula $\operatornamewithlimits{vol}(B^{n}_{1,1})=2^{n}/n!$. The application of the polarization identity to other values of $q\not=1$ is also possible, but one arrives to an $n$-dimensional integral, which (in contrary to $I_{p}^{n}$) seems to be hard to compute explicitly.

3 Asymptotic behavior

Volumes of convex and non-convex bodies play an important role in many areas of mathematics, cf. [39]. Nevertheless, for most of the applications we do not need the exact information about the volume, it is often enough to apply good lower and/or upper bounds of this quantity. For example, for the use in local Banach theory, it is sometimes sufficient to have some asymptotic bounds on $\operatornamewithlimits{vol}(B_{p}^{n})$ for $n$ large. In this section, we provide two such estimates.

3.1 Asymptotic behavior of $\operatornamewithlimits{vol}(B_{p,q}^{n})^{1/n}$

The first quantity, we would like to study, is the $n$-th root of $\operatornamewithlimits{vol}(B^{n}_{p,q})$. In the Lebesgue case $q=p$, (4) can be combined with the Stirling’s formula (cf. [44])

[TABLE]

where $0<\theta(t)<1/12$ for all $t>0$, to show that

[TABLE]

where the constants of equivalence do not depend on $n$. Combining (19) with the embedding (cf. Theorem 8)

[TABLE]

we observe that

[TABLE]

for all $0<p\leq\infty.$ The aim of this section is to show, that the lower bound in (20) is sharp and that (19) generalizes to all $0<p<\infty$ and $0<q\leq\infty$ without additional logarithmic factors.

If $0<p<\infty$ and $q=1$, this can be obtained as a consequence of Theorem 5. Indeed, elementary estimates give

[TABLE]

with constants independent on $k$ and Theorem 5 then implies

[TABLE]

The result is then finished by another application of Stirling’s formula.

To extend the result also to $q\not=1$, we apply the technique of entropy together with interpolation.

Theorem 6**.**

Let $n\in\mathbb{N}$, $0<p<\infty$ and $0<q\leq\infty$. Then

[TABLE]

with the constants of equivalence independent of $n$.

Proof.

Step 1.: First, we show the upper bound of $\operatornamewithlimits{vol}(B^{n}_{p,\infty})^{1/n}$. For that reason, we define the entropy numbers of a bounded linear operator between two quasi-normed Banach spaces $X$ and $Y$ as follows

[TABLE]

Here, $B_{X}$ and $B_{Y}$ stand for the unit ball of $X$ and $Y$, respectively.

We use the interpolation inequality for entropy numbers (cf. Theorem [15, Theorem 1.3.2 (i)]) together with the interpolation property of Lorentz spaces (cf. [5, Theorems 5.2.1 and 5.3.1]) and obtain that

[TABLE]

where $c_{p}>0$ depends only on $p$. Together with the known estimates of entropy numbers of embeddings of Lebesgue-type sequence spaces [21, 26, 27, 41], we obtain

[TABLE]

By the definition of entropy numbers, this means that $B_{p,\infty}^{n}$ can be covered with $2^{n-1}$ balls in $\ell_{\infty}^{n}$ with radius $(1+\varepsilon)c_{p}n^{-1/p}$ for every $\varepsilon>0$. Comparing the volumes, we obtain

[TABLE]

i.e. $\operatornamewithlimits{vol}(B_{p,\infty}^{n})^{1/n}\leq c_{p}^{\prime}n^{-1/p}.$

Step 2.: The estimate from above for general $0<q\leq\infty$ is covered by the embedding of Theorem 8 and by the previous step.

Step 3.: For the lower bound, we use again the interpolation of entropy numbers leading to

[TABLE]

Therefore, the unit ball $B_{p/2}^{n}$ can be covered by $2^{n-1}$ copies of $B_{p,q}^{n}$ multiplied by $(1+\varepsilon)cn^{-1/p}$. Comparing the volumes, we obtain

[TABLE]

which finishes the proof. ∎

The result of Theorem 6 seems to be a bit surprising at first look - especially in view of (20), which suggests that some additional logarithmic factor could appear. That the outcome of Theorem 6 was by no means obvious is confirmed by inspecting the case $p=\infty$, where the behavior of $n$-th root of the volume of the unit ball actually differs from (21).

Theorem 7**.**

Let $n\in\mathbb{N}$ be a positive integer. Then

[TABLE]

with the constants of equivalence independent on $n$.

Proof.

By Theorem 5, we know that

[TABLE]

where

[TABLE]

Therefore,

[TABLE]

The lower bound of this quantity is straightforward

[TABLE]

For the upper bound, we use the inequality between geometric and arithmetic mean and obtain

[TABLE]

The last integral is known as the (offset) logarithmic integral and is known to be asymptotically $O(x/\log(x))$ for $x$ going to infinity, cf. [1, Chapter 5]. Alternatively, the same fact can be shown easily by the L’Hospital’s rule. This finishes the proof. ∎

3.2 Ratio of volumes

The unit balls $B^{n}_{p,\infty}$ of weak Lebesgue spaces are commonly considered to be “slightly larger” than the unit balls of Lebesgue spaces with the same summability parameter. The aim of this section is to study their relation in more detail. For that sake, we define for $0<p<\infty$

[TABLE]

By the embedding in Theorem 8 (which we give below with the full proof for reader’s convenience, cf. [4, Chapter 4, Proposition 4.2]) we know that this quantity is bounded from below by one. Later on, we would like to study its behavior (i.e. growth) when $n$ tends to $\infty.$

Theorem 8**.**

If $0<p<\infty$ and $0<q\leq r\leq\infty$, then

[TABLE]

where the quantity $c_{p,q,r}$ does not depend on $n$. In particular, $B_{p,q}^{n}\subset B_{p,r}^{n}$ if also $q\leq p$.

Proof.

First, we prove the assertion with $r=\infty$. If $1\leq l\leq n$ is a positive integer, the result follows from

[TABLE]

and

[TABLE]

We obtain that

[TABLE]

For $q\leq p$, it can be shown by elementary calculus that

[TABLE]

Together with (25), this implies that $\|x\|_{p,\infty}\leq\|x\|_{p,q}$ and we obtain $B_{p,q}^{n}\subset B_{p,\infty}^{n},$ i.e. (24) with $c_{p,q,\infty}=1.$ If, on the other hand, $q>p$ we estimate

[TABLE]

and (24) follows with $c_{p,q,\infty}=(q/p)^{1/q}$.

If $0<q<r<\infty$, we write

[TABLE]

i.e.

[TABLE]

∎

We show that the ratio $R_{p,n}$ defined in (23) grows exponentially for $0<p\leq 2$. Naturally, we also conjecture that the same is true for all $0<p<\infty$, but we leave this as an open problem.

Theorem 9**.**

For every $0<p\leq 2$, there is a constant $C_{p}>1$, such that

[TABLE]

with the multiplicative constant independent on $n$.

Proof.

We give the proof for even $n$’s, the proof for odd $n$’s is similar, only slightly more technical. Let ${\mathcal{B}^{n}_{p}}\subset\mathbb{R}^{n}$ be the set of vectors $x\in\mathbb{R}^{n}$, which satisfy

[TABLE]

and \displaystyle x^{*}_{n/2+1},\dots,x_{n}^{*}\in\Bigl{[}0,\frac{1}{n^{1/p}}\Bigr{]}. Then ${\mathcal{B}}_{p}^{n}\subset B_{p,\infty}^{n}$ and the volume of ${\mathcal{B}}_{p}^{n}$ can be calculated by combinatorial methods. Indeed, there is ${n\choose n/2}$ ways how to choose the $n/2$ indices of the smallest coordinates. Furthermore, there is $(n/2)!$ ways, how to distribute the $n/2$ largest coordinates. We obtain that

[TABLE]

First, we observe that, by Stirling’s formula (18),

[TABLE]

By the mean value theorem, we obtain

[TABLE]

We use (18) to estimate $\Gamma(1+1/p)$ together with (26) and (27) and obtain

[TABLE]

where $\alpha=0$ for $0<p\leq 1$ and $\alpha=1$ for $1<p\leq 2.$ The proof is then finished by monotonicity and

[TABLE]

∎

4 Entropy numbers

We have already seen the closed connection between estimates of volumes of unit balls of finitedimensional (quasi-)Banach spaces and the decay of entropy numbers of embeddings of such spaces in the proof of Theorem 6. With the same arguments as there, it is rather straightforward to prove that

[TABLE]

for $0<p_{0},p_{1}<\infty$ with $p_{0}\not=p_{1}.$ On the other hand, it was shown in [14], that the entropy numbers of diagonal operators between Lorentz sequence spaces can exhibit also a very complicated behavior. Actually, they served in [14] as the first counterexample to a commonly conjectured interpolation inequality for entropy numbers.

We complement (28) by considering the limiting case $p_{0}=p_{1}.$ As an application of our volume estimates, accompanied by further arguments, we will investigate in this section the decay of the entropy numbers $e_{k}(id:\ell^{n}_{1,\infty}\to\ell^{n}_{1}).$

Before we come to our main result, we state a result from coding theory [18, 36], which turned out to be useful also in connection with entropy numbers [14, 26] and even in optimality of sparse recovery in compressed sensing [6, 16, 17].

Lemma 10**.**

Let $k\leq n$ be positive integers. Then there are $M$ subsets $T_{1},\dots,T_{M}$ of $\{1,\dots,n\}$, such that

1. (i)

\displaystyle M\geq\Bigl{(}\frac{n}{4k}\Bigr{)}^{k/2}, 2. (ii)

$|T_{i}|=k$* for all $k=1,\dots,M$,* 3. (iii)

$|T_{i}\cap T_{j}|<k/2$* for all $i\not=j.$*

To keep the argument simple, we restrict ourselves to $p=1$.

Theorem 11**.**

Let $k$ and $n$ be positive integers. Then

[TABLE]

where the constants of equivalence do not depend on $k$ and $n$.

Proof.

Step 1. (lower bound for $k\geq n$): If $B_{1,\infty}^{n}$ is covered by $2^{k-1}$ balls in $\ell_{1}^{n}$ with radius $\varepsilon>0$, it must hold

[TABLE]

which (in combination with Theorem 6) gives the lower bound for $k\geq n$.

Step 2. (upper bound for $k\geq n$): We use again volume arguments. Let $\varepsilon>0$ be a parameter to be chosen later on. Let $\{x_{1},\dots,x_{N}\}\subset B_{1,\infty}^{n}$ be a maximal $\varepsilon$-distant set in the metric of $\ell_{1}^{n}.$ This means that

[TABLE]

and $\|x_{i}-x_{j}\|_{1}>\varepsilon$ for $i\not=j.$ Hence, any time $N\leq 2^{k-1}$ for some positive integer $k$, then $e_{k}(id:\ell_{1,\infty}^{n}\to\ell_{1}^{n})\leq\varepsilon.$ To estimate $N$ from above, let us note that $(x_{j}+\varepsilon B_{1}^{n})\subset 2(1+\varepsilon)B_{1,\infty}^{n}$, which follows by the quasi-triangle inequality for $\ell_{1,\infty}^{n}$. On the other hand, the triangle inequality of $\ell_{1}^{n}$ implies that $(x_{j}+\frac{\varepsilon}{2}B_{1}^{n})$ are mutually disjoint. Hence,

[TABLE]

i.e.

[TABLE]

if $0<\varepsilon<1.$ We now define the parameter $\varepsilon$ by setting the right-hand side of (29) equal to $2^{k-1}$. By Theorem 6, there exists an integer $\gamma\geq 1$, such that $\varepsilon<1$ for $k\geq\gamma n$. In this way, we get $N\leq 2^{k-1}$ and $\varepsilon=8[\operatornamewithlimits{vol}(B_{1,\infty}^{n})/\operatornamewithlimits{vol}(B_{1}^{n})]^{1/n}\cdot 2^{-\frac{k-1}{n}}\leq c\,2^{-\frac{k-1}{n}}$. This gives the result for $k\geq\gamma n.$

Step 3. (upper bound for $k\leq n$): Let $1\leq l\leq n/2$ be a positive integer, which we will chose later on. To every $x\in B_{1,\infty}^{n}$, we associate $S\subset\{1,\dots,n\}$ to be the indices of its $l$ largest entries (in absolute value). Furthermore, $x_{S}\in\mathbb{R}^{n}$ denotes the restriction of $x$ to $S$. We know that

[TABLE]

By Step 2, there is an absolute constant $c>0$ (independent of $l$), such that

[TABLE]

where $\gamma\geq 1$ is the integer constant from Step 2.

Hence, there is a point set ${\mathcal{N}}\subset\mathbb{R}^{l}$, with $|{\mathcal{N}}|=2^{\gamma l}$, which is a $c$-net of $B_{1,\infty}^{l}$ in the $\ell_{1}^{l}$-norm. For any set $S$ as above, we embed ${\mathcal{N}}$ into $\mathbb{R}^{n}$ by extending the points from ${\mathcal{N}}$ by zero outside of $S$ and obtain a point set ${\mathcal{N}}_{S}$, which is a $c$-net of $\{x\in B_{1,\infty}^{n}:{\rm supp\ }(x)\subset S\}$. Taking the union of all these nets over all sets $S\subset\{1,\dots,n\}$ with $|S|=l$, we get $2^{\gamma l}{n\choose l}$ points, which can approximate any $x\in B_{1,\infty}^{n}$ within $c+\log(n/l)$ in the $\ell_{1}^{n}$-norm.

We use the elementary estimate ${n\choose l}\leq(en/l)^{l}$ and assume (without loss of generality) that $\gamma\geq 2$. Then we may conclude, that whenever $2^{k-1}\geq 2^{\gamma l}{n\choose l}$, we have $e_{k}(id:\ell_{1,\infty}^{n}\to\ell_{1}^{n})\leq c+\log(n/l)$, i.e.

[TABLE]

By a standard technical argument, $l$ can be chosen up to the order of $k/\log(n/k)$, which gives the result for $n$ large enough and $k$ between $(\gamma+1)\log(en)$ and $n$. Using monotonicity of entropy numbers, the upper bound from Step 2 and the elementary bound $e_{k}(id:\ell_{1,\infty}^{n}\to\ell_{1}^{n})\leq\|id:\ell_{1,\infty}^{n}\to\ell_{1}^{n}\|\leq 1+\log(n)$ concludes the proof of the upper bounds.

Step 4. (lower bound for $k\leq n$): Let $n$ be a sufficiently large positive integer and let $\nu\geq 1$ be the largest integer with $12\cdot 4^{\nu}\leq n$. Let $1\leq\mu\leq\nu$ be a positive integer.

We apply Lemma 10 with $k$ replaced by $4^{l}$ for every integer $l$ with $\mu\leq l\leq\nu$. In this way, we obtain a system of subsets $T^{l}_{1},\dots,T^{l}_{M_{l}}$ of $\{1,\dots,n\}$, such that $|T^{l}_{i}|=4^{l}$ for every $1\leq i\leq M_{l}$, $|T^{l}_{i}\cap T^{l}_{j}|<4^{l}/2$ for $i\not=j$ and

[TABLE]

For $j\in\{1,\dots,M\}$, we put

[TABLE]

Observe, that by this construction the sets $\{\widetilde{T}^{l}_{j}:\mu\leq l\leq\nu\}$ are mutually disjoint and $|\widetilde{T}^{l}_{j}|\leq|T^{l}_{j}|=4^{l}$. Furthermore, $|\widetilde{T}^{\mu}_{j}|=4^{\mu}$ and

[TABLE]

for $\mu<l\leq\nu.$

We associate to the sets $\{\widetilde{T}_{j}^{l}:\mu\leq l\leq\nu,1\leq j\leq M\}$ a system of vectors $x^{1},\dots,x^{M}\in{\mathbb{R}}^{n}$. First, we observe that if $u\in\{1,\dots,n\}$ belongs to $\widetilde{T}^{l}_{j}$ for some $l\in\{\mu,\mu+1,\dots,\nu\}$, then this $l$ is unique and we put $(x^{j})_{u}=\frac{1}{4^{l}}.$ Otherwise, we set $(x^{j})_{u}=0.$ We may also express this construction by

[TABLE]

where $\chi_{A}$ is the indicator function of a set $A$.

Now we observe that

[TABLE]

Furthermore, let $i\not=j$ and let $u\in\widetilde{T}^{l}_{i}$ with $u\not\in\widetilde{T}^{l}_{j}$. Then

[TABLE]

To estimate the $\ell_{1}$-distances among the points $\{x^{1},\dots,x^{M}\}$, we combine (31), (30), and obtain for $i\not=j$

[TABLE]

We conclude, that the points $\{x^{j}:j=1,\dots,M\}$ satisfy

[TABLE]

It follows that if a positive integer $k$ satisfies

[TABLE]

then

[TABLE]

where the absolute constant $c$ can be taken $c=\frac{3}{64}.$

Let now $n\geq 200$ and $1\leq k\leq n/200$ be positive integers. Then we chose $\nu\geq 1$ to be the largest integer with $12\cdot 4^{\nu}\leq n$ and let $\mu\geq 1$ be the smallest integer with $k\leq 4^{\mu}/2.$ Due to $\frac{n}{4^{\mu+1}}\geq 2$, this choice ensures (32) and

[TABLE]

The remaining pairs of $k$ and $n$ are covered by monotonicity of entropy numbers at the cost of constants of equivalence. ∎

Acknowledgement: We would like to thank Franck Barthe for proposing the problem to us and to Leonardo Colzani and Henning Kempka for valuable discussions.