An analytic approach to the Ubiquity of geometric Brascamp–Lieb data

Abstract.

The ubiquity of geometric Brascamp–Lieb data, which means a certain kind of density of geometric data in the set of all feasible Brascamp–Lieb data has been studied recently. Relying substantially on the work of Dvir and Hu, we provide an analytic proof of ubiquity. Our argument also extends to the setting of quiver Brascamp–Lieb data.

1. Introduction

In their research on the adjoint Brascamp–Lieb inequality, Bennett and Tao showed that the finiteness of the adjoint Brascamp–Lieb constant is equivalent to the finiteness of the Brascamp–Lieb constant [6]. More recently, using density-type considerations, a new proof of part of this equivalence was given in [8]. This argument was based on the fact that so-called geometric data are dense (in a suitable sense) in the space of all feasible data. Geometric data are relatively tractable, and such density results, refered to as ubiquity in [8], are expected to be useful in reducing certain proofs concerning feasible data to the case of geometric data. In this paper we use ideas from work by Dvir–Hu [12] to give a new proof of ubiquity of geometric Brascamp–Lieb data. In fact, our proof also works in the more general context of quiver Brascamp–Lieb data and so we also recover a very recent result by Chindris–Derksen [11].

1.1. Brascamp–Lieb inequality

Let us begin by introducing the form of the Brascamp–Lieb inequality.

Given a family of linear transformations $\mathbf{B}=(B_{j}\in\mathbb{R}^{n_{j}\times{n}})_{j\in[m]}$ and a tuple of positive exponents $\mathbf{p}=(p_{j})_{j\in[m]}$, a Brascamp–Lieb ineqality is an inequality of the form

(1.1) $$\int_{\mathbb{R}^{n}}\prod_{j=1}^{m}f_{j}(B_{j}x)^{p_{j}}\,dx\leq C\prod_{j=1}^{m}\left(\int_{\mathbb{R}^{n_{j}}}f_{j}\right)^{p_{j}}$$

which holds for all non-negative and integrable functions $f_{j}:\mathbb{R}^{n_{j}}\to\mathbb{R}$. We denote by $\mathrm{BL}({\mathbf{B}},\mathbf{p})$ the optimal constant in (1.1), and by $\mathrm{BL_{g}}({\mathbf{B}},\mathbf{p})$ the optimal constant in (1.1) when $f_{j}(x):=\exp{(-\pi\langle A_{j}x,x\rangle)}$ for some $A_{j}\in\mathsf{PD}_{n_{j}}$. Here $\mathsf{PD}_{n_{j}}$ denotes the set of all real, self-adjoint and positive definite $n_{j}\times n_{j}$ matrices.

Hölder’s inequality, the Loomis–Whitney inequality and Young’s convolution inequality are important special cases. The Brascamp–Lieb inequality has been studied by many authors, and some useful results about the finiteness and the behaviour of the constant $\mathrm{BL_{g}}({\mathbf{B}},\mathbf{p})$ and $\mathrm{BL}({\mathbf{B}},\mathbf{p})$ have been obtained (see, for example [2], [4], [7] and [5]). A remarkable theorem of Lieb [16] gives us the following.

Theorem 1.1.

Let $(\mathbf{B},\mathbf{p})$ be a Brascamp–Lieb datum. Then $\mathrm{BL}(\mathbf{B},\mathbf{p})=\mathrm{BL_{g}}(\mathbf{B},\mathbf{p})$.

We next introduce geometric Brascamp–Lieb data, a special case of Brascamp–Lieb data.

Definition 1.2.

A Brascamp–Lieb datum $({\mathbf{B}},\mathbf{p})$ is called geometric if

(1.2) $$\sum^{m}_{j=1}p_{j}B_{j}^{T}B_{j}=id_{{\mathbb{R}^{n}}}$$

and

(1.3) $$B_{j}B_{j}^{T}=id_{{\mathbb{R}^{n_{j}}}}\quad\text{for all $j\in[m]$}.$$

Theorem 1.3.

Let $({\mathbf{B}},\mathbf{p})$ be a geometric Brascamp–Lieb datum. Then

$$\mathrm{BL}({\mathbf{B}},\mathbf{p})=\mathrm{BL_{g}}({\mathbf{B}},\mathbf{p})=1.$$

Theorem 1.3 is first proved by Ball (rank one) [1] and Barthe (general rank) [2]. (See also [5, Theorem 2.8].) In general, it is difficult to compute $\mathrm{BL}({\mathbf{B}},\mathbf{p})$. So, geometric data are easier to handle in that respect.

We say that the two tuples of matrices $\mathbf{B}$ and $\mathbf{B}^{\prime}$ are equivalent if there exist invertible matrices $(C_{j}\in\mathbb{R}^{n_{j}\times n_{j}})_{j\in[m]}$ and $C\in\mathbb{R}^{n\times n}$ such that for each $j\in[m]$,

$$B^{\prime}_{j}=C_{j}^{-1}B_{j}C.$$

According to [5, Lemma 3.3], we have

(1.4) $$\mathrm{BL}({\mathbf{B}^{\prime}},\mathbf{p})=\frac{\prod^{m}_{j=1}|\mathrm{det_{\mathbb{R}^{n_{j}}}(\textit{$C_{j}$})}|^{p_{j}}}{|\mathrm{det_{\mathbb{R}^{n}}(\textit{$C$})}|}\mathrm{BL}({\mathbf{B}},\mathbf{p}).$$

By the following theorem [8], which is a density-type result, we may expect problems involving Brascamp–Lieb data with $\mathrm{BL}({\mathbf{B}},\mathbf{p})<\infty$, which we refer to as feasible, can be reduced to the case of geometric data. Bez, Gauvan and Tsuji proved this theorem using operator scalings [14]. Roughly speaking, operator scaling is an iterative normalisation procedure that rescales the linear maps so that they approach a geometric datum.

Theorem 1.4.

Let $\mathbf{p}=(p_{j})_{j\in[m]}$ be a tuple of positive real numbers. Then for any feasible datum $(\mathbf{B},\mathbf{p})$, there exists some tuple of matrices $\mathbf{G}$ such that $(\mathbf{G},\mathbf{p})$ is geometric and for any $\varepsilon>0$, there is some tuple of matrices $\mathbf{B}^{\prime}$ which is equivalent to $\mathbf{B}$ such that

$$\|\mathbf{G}-\mathbf{B}^{\prime}\|<\varepsilon.$$

Recently, the notion of Brascamp–Lieb datum has been extended to so-called quiver datum by Chindris and Derksen [11].

1.2. Capacity

Let $k$ and $m$ be positive integers. Let $(d_{1},\ldots,d_{k})$ and $(n_{1},\ldots,n_{m})$ be two tuples of positive integers and $\mathbf{p}=(p_{1},\ldots,p_{m})$ be a tuple of positive real numbers such that

$$\sum_{i\in[k]}d_{i}=\sum_{j\in[m]}p_{j}n_{j}.$$

For each pair $(i,j)$, let $\mathbf{A}_{ij}$ be a finite index set and denote

$$\mathbf{V}:=(V_{a}\in{\mathbb{R}}^{n_{j}\times d_{i}}:a\in{\mathbf{A}}_{ij},i\in[k],j\in[m]).$$

The pair $(\mathbf{V},\mathbf{p})$ is called a quiver datum, and its capacity is defined as follows:

(1.5) $$\mathrm{Cap}(\mathbf{V},\mathbf{p}):=\inf_{(Y_{j}\in\mathsf{PD}_{n_{j}})_{j\in[m]}}\mathrm{Cap}(\mathbf{V},\mathbf{p};(Y_{j})_{j\in[m]}),$$

where

(1.6) $$\mathrm{Cap}(\mathbf{V},\mathbf{p};(Y_{j})_{j\in[m]}):=\frac{\prod^{k}_{i=1}\det_{{\mathbb{R}}^{d_{i}}}\big(\sum^{m}_{j=1}p_{j}\sum_{a\in\mathbf{A}_{ij}}V^{T}_{a}Y_{j}V_{a}\big)}{\prod^{n}_{j=1}(\det_{{\mathbb{R}}^{n_{j}}}Y_{j})^{p_{i}}}.$$

Considering the special case $k=1$ and $\#\mathbf{A}_{1j}=1$ for all $j\in[m]$, we have

$$\mathrm{Cap(\mathbf{B},\mathbf{p})=\mathrm{BL_{g}}(\mathbf{B},\mathbf{p})^{-2}}.$$

Thus, by Lieb’s theorem, we can study the finiteness and the behaviour of the constant $\mathrm{BL}(\mathbf{B},\mathbf{p})$ by studying the associated capacity. Chindris and Derksen have developed the theory of the capacity. Their work [10] gives us important basic properties, following the ordinary theory of the Brascamp–Lieb inequality. Also, [11] gives us the algebraicity of the Brascamp–Lieb constant in the case where the exponents are rational. From here, we introduce results regarding the positivity and extremals of capacity, as well as geometricity of quiver data.

We say that $(\mathbf{V},\mathbf{p})$ is feasible when its capacity is strictly positive. Also, we say that the two tuples of matrices $\mathbf{V}$, $\mathbf{V}^{\prime}$ are equivalent if there exist some tuples of invertible linear maps $(M_{i}\in{\mathbb{R}}^{d_{i}\times d_{i}})_{i\in[k]}$ and $(N_{j}\in{\mathbb{R}}^{n_{j}\times n_{j}})_{j\in[m]}$ such that

$$V_{a}^{\prime}=N_{j}V_{a}M_{i}^{-1}\quad\text{for all $a\in\mathbf{A}_{ij}$, $i\in[k]$, $j\in[m]$}.$$

From the calculation in [10, Theorem 14], we obtain the following property.

Proposition 1.5.

Assume that the two tuples of matrices $\mathbf{V}$, $\mathbf{V}^{\prime}$ are equivalent. Then $(\mathbf{V},\mathbf{p})$ is feasible if and only if $(\mathbf{V}^{\prime},\mathbf{p})$ is feasible.

We say that $(\mathbf{V},\mathbf{p})$ is extremisable if there is some $(Y_{j}\in\mathsf{PD}_{n_{j}})_{j\in[m]}$ achieving the infimum on the right-hand side of (1.5), and we call this $(Y_{j}\in\mathsf{PD}_{n_{j}})_{j\in[m]}$ an extremiser of $(\mathbf{V},\mathbf{p})$. The following fact about extremisers is known (see [11, Theorem 5]).

Theorem 1.6.

Assume that $(\mathbf{V},\mathbf{p})$ is extremisable. Then $(Y_{j}\in\mathsf{PD}_{n_{j}})_{j\in[m]}$ is an extremiser of $(\mathbf{V},\mathbf{p})$ if and only if $(Y_{j}\in\mathsf{PD}_{n_{j}})_{j\in[m]}$ satisfies the following property:

(1.7) $$X_{i}:=\sum^{m}_{j=1}p_{j}\sum_{a\in{\mathbf{A}}_{ij}}(V_{a})^{T}Y_{j}V_{a}\quad\text{is invertible for all $i\in[k]$ }$$

and

(1.8) $$\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}V_{a}{X_{i}}^{-1}(V_{a})^{T}={Y_{j}}^{-1}\quad\text{for all $j\in[m]$}.$$

The fact that (1.7) and (1.8) are necessary for the existence of extremisers is proved in [10, Theorem 20], using the same method introduced in [5]. Also, the converse is proved in [11, Theorem 5].

In addition to the special cases of the quiver datum introduced already, we next introduce the notion of geometric quiver datum.

Definition 1.7.

The quiver datum $(\mathbf{V},\mathbf{p})$ is said to be geometric if

$$\sum^{m}_{j=1}p_{j}\sum_{a\in{\mathbf{A}}_{ij}}(V_{a})^{T}V_{a}=id_{{\mathbb{R}}^{d_{i}}}\quad\text{for each $i\in[k]$ }$$

and

$$\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}V_{a}(V_{a})^{T}=id_{{\mathbb{R}}^{n_{j}}}\quad\text{for each $j\in[m]$}.$$

Chindris and Derksen proved that the capacity of a geometric datum is equal to one [11, Theorem 3].

Theorem 1.8.

If $(\mathbf{V},\mathbf{p})$ is a geometric quiver datum, then $\mathrm{Cap}(\mathbf{V},\mathbf{p})=1$.

In [11], Chindris and Derksen also proved an extension of Theorem 1.4 to quiver data. They proved it by algebraic methods via a Jordan–Holder filtration.

Theorem 1.9.

Let $\mathbf{p}=(p_{j})_{j\in[m]}$ be a tuple of positive real numbers. Then for any feasible quiver datum $(\mathbf{V},\mathbf{p})$, there exists some tuple of matrices $\mathbf{G}$ such that $(\mathbf{G},\mathbf{p})$ is geometric and for any $\varepsilon>0$, there is some tuple of matrices $\mathbf{V}^{\prime}$ which is equivalent to $\mathbf{V}$ such that

$$\|\mathbf{G}-\mathbf{V}^{\prime}\|<\varepsilon.$$

Unlike the proof of ubiquity by Bez, Gauvan, and Tsuji using operator scalings, or the proof by Chindris and Derksen based on algebraic methods, this paper provides a completely new proof of Theorem 1.9 using analytic techniques. With the goal of establishing ubiquity, we prove the following theorem.

Theorem 1.10.

Assume that the quiver datum $(\mathbf{V},\mathbf{p})$ is feasible. Then for each $\varepsilon>0$, there exists some quiver datum $\mathbf{V}^{\prime}=(V_{a}^{\prime})_{a}$ equivalent to $\mathbf{V}$ such that

(1.9) $$\sum^{m}_{j=1}p_{j}\sum_{a\in{\mathbf{A}}_{ij}}(V_{a}^{\prime})^{T}V_{a}^{\prime}=id_{{\mathbb{R}}^{d_{i}}}\quad\text{for each $i\in[k]$ }$$

and

(1.10) $$\Big\|\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}V_{a}^{\prime}(V_{a}^{\prime})^{T}-id_{{\mathbb{R}}^{n_{j}}}\Big\|<\varepsilon\quad\text{for each $j\in[m]$ }.$$

In Section 2, we study the logarithm of capacity and prove Theorem 1.10. The proof is based on the work of Dvir and Hu [12].

In Section 3, we show that (1.7) and (1.8) of Theorem 1.6 are necessary for the existence of extremisers using the proof of Theorem 1.10. In addition, we prove Theorem 1.9 using the result of Theorem 1.10. After that, using ubiquity, we provide an upper bound for Brascamp–Lieb constants for certain data.

2. Proof of Theorem 1.10

Guided by the proof of [12, Theorem 1.2], we now proceed to prove Theorem 1.10.

2.1. Preliminaries

First, we reduce the positivity of the capacity to the boundedness of a function. Similar reductions are carried out in [2, Proposition 6].

For each $j\in[m]$ and $Y_{j}\in\mathsf{PD}_{n_{j}}$, there exists some orthogonal matrix $R_{j}\in O_{n_{j}}(\mathbb{R})$ and $t_{j1},\ldots,t_{jn_{j}}\in\mathbb{R}$ such that

$$Y_{j}=R_{j}\begin{pmatrix}\exp{{t_{j1}}}\ &&\text{\huge{0}}\\ &\ddots&\\ \text{\huge{0}}&&\exp{t_{jn_{j}}}\end{pmatrix}R_{j}^{T}.$$

The middle matrix is a diagonal matrix with $\exp{{t_{js}}}$ as its $(s,s)$ entry. For each $j\in[m]$, let $[{\bm{x}}_{a,j1},\ldots,{\bm{x}}_{a,jn_{j}}]$ be a representation of the matrix $V_{a}^{T}R_{j}$ as a list of column vectors. Let $N=\sum_{j\in[m]}n_{j}$ and define $f:\mathbb{R}^{N}\times\prod^{m}_{j=1}O_{n_{j}}(\mathbb{R})\to\mathbb{R}$ as follows:

$$f(\bm{t},R_{1},\ldots,R_{m}):=\langle\bm{\gamma},\bm{t}\rangle-\sum^{k}_{i=1}\log\det X_{i},$$

where $\bm{\gamma}=(\gamma_{e})_{e\in[N]}=(\gamma_{js})_{j\in[m],s\in[n_{j}]}$, $\gamma_{js}=p_{j}$ for each $j\in[m],s\in[n_{j}]$ and

$$X_{i} =\sum_{j\in[m]}p_{j}\sum_{a\in{\mathbf{A}}_{ij}}\sum_{s\in[n_{j}]}\exp{t_{js}}\cdot{\bm{x}}_{a,js}{\bm{x}}^{T}_{a,js}$$

$$=\sum_{j\in[m]}p_{j}\sum_{a\in{\mathbf{A}}_{ij}}[{\bm{x}}_{a,j1},\ldots,{\bm{x}}_{a,jn_{j}}]\begin{pmatrix}\exp{{t_{j1}}}\ &&\text{\huge{0}}\\ &\ddots&\\ \text{\huge{0}}&&\exp{t_{jn_{j}}}\end{pmatrix}\begin{pmatrix}{\bm{x}}_{a,j1}^{T}\\ :\\ :\\ {\bm{x}}_{a,jn_{j}}^{T}\end{pmatrix}.$$

Observe that $X_{i}$ is self-adjoint and positive semi-definite. Taking the logarithm of (1.6), we obtain

$$f(\bm{t},R_{1},\ldots,R_{m})=-\log\mathrm{Cap}(\mathbf{V},\mathbf{p};(Y_{j})_{j\in[m]}).$$

Thus, we obtain the following proposition.

Proposition 2.1.

The function $f$ is bounded above if and only if $(\mathbf{V},\mathbf{p})$ is feasible.

From Proposition 2.1 we may assume that the function $f$ defined earlier is bounded above. In particular, every $X_{i}$ is always positive definite. This means that there exists an invertible matrix $M_{i}$ such that

$$M_{i}^{T}M_{i}=X_{i}^{-1}$$

for each $i\in[k]$.

2.2. Basic properties of the function $f$

We use the following four lemmas to prove Theorem 1.10. The idea of the proof is similar to [12] (see also [13], [2]), which considers a sequence approaching the extreme point of the function.

Lemma 2.2.

For any ${\bm{t}}\in\mathbb{R}^{N}$, there exist orthogonal triples $(R^{*}_{1}(\bm{t}),\ldots,R^{*}_{m}(\bm{t}))$ such that

(2.1) $$f(\bm{t},R^{*}_{1}({\bm{t}}),\ldots,R^{*}_{m}({\bm{t}}))=\max_{R_{1},\ldots,R_{m}}f(\bm{t},R_{1},\ldots,R_{m})$$

and for each $j\in[m]$ and $s,s^{\prime}\in[n_{j}]$, $s\neq s^{\prime}$ with $t_{js}=t_{js^{\prime}}$,

(2.2) $$\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\langle{M_{i}{\bm{x}}_{a,js}},{M_{i}{\bm{x}}_{a,js^{\prime}}}\rangle=0$$

where $[{\bm{x}}_{a,i1},\ldots,{\bm{x}}_{a,jn_{j}}]=V_{a}^{T}R^{*}_{i}({\bm{t}})$.

Proof.

First, there are some orthogonal triples $(R^{*}_{1}({\bm{t}}),\ldots,R^{*}_{m}({\bm{t}}))$ satisfying (2.1) by the compactness of $\prod^{m}_{j=1}O_{n_{j}}(\mathbb{R})$.

Fix $j\in[m]$. We partition the indices of $(t_{j1},\ldots,t_{jn_{j}})$ into equivalence classes $J_{1},\ldots,J_{b}\subset[n_{j}]$ such that for $s$, $s^{\prime}$ in the same classes $t_{js}=t_{js^{\prime}}$ and for different classes $t_{js}\neq t_{js^{\prime}}$. We use $t_{S_{r}}$ to denote the value of $t_{js}$ for $s\in J_{r}$, and $L_{a,J_{r}}$ to denote the matrix consisting of all columns $\bm{x}_{a,js}$ with $s\in J_{r}$. Since the terms in $X_{i}$ that depend on $R_{j}$ are

$$\sum_{a\in{\mathbf{A}}_{ij}}\sum_{r\in[b]}\Big(p_{j}\exp{t_{J_{r}}}\sum_{s\in J_{r}}{\bm{x}_{a,js}{\bm{x}}^{T}_{a,js}}\Big)=\sum_{a\in{\mathbf{A}}_{ij}}\sum_{r\in[b]}\big(p_{j}\exp{t_{J_{r}}}\cdot L_{a,J_{r}}L^{T}_{a,J_{r}}\big)$$

$$=\sum_{a\in{\mathbf{A}}_{ij}}\sum_{r\in[b]}\big(p_{j}\exp{t_{J_{r}}}\cdot L_{a,J_{r}}Q_{r}Q^{T}_{r}L^{T}_{a,J_{r}}\big),$$

where $Q_{r}\in O_{J_{r}}(\mathbb{R})$ is independent of $a\in{\mathbf{A}}_{ij}$ and $i\in[k]$, we can replace $R^{*}_{i}({\bm{t}})$ with $R^{*}_{i}({\bm{t}})\mathrm{diag}(Q_{1},\ldots,Q_{b})$ without changing the value of $X_{i}$, $M_{i}$ and $f$.

For each $r\in[b]$, $\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}(M_{i}L_{a,J_{r}})^{T}(M_{i}L_{a,J_{r}})$ is a real self-adjoint matrix, so there exists a $|J_{r}|\times|J_{r}|$ orthogonal matrix $Q_{r}$ such that

$$Q_{r}^{T}\Big(\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}( M_{i}L_{a,J_{r}})^{T}(M_{i}L_{a,J_{r}})\Big)Q_{r}$$

$$=\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}(M_{i}L_{a,J_{r}}Q_{r})^{T}(M_{i}L_{a,J_{r}}Q_{r})$$

$$=\Big(\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\langle{M_{i}\tilde{\bm{x}}_{a,js}},{M_{i}\tilde{\bm{x}}_{a,js^{\prime}}}\rangle\Big)_{s,s^{\prime}}\in M_{|J_{r}|\times|J_{r}|}(\mathbb{R})$$

is diagonal, where

$$[\tilde{\bm{x}}_{a,j1},\ldots,\tilde{\bm{x}}_{a,j|J_{r}|}]=L_{a,J_{r}}Q_{r}.$$

Thus, by replacing $R^{*}_{j}({\bm{t}})$ with $R^{*}_{j}({\bm{t}})\mathrm{diag}(Q_{1},\ldots,Q_{b})$ (here $R^{*}_{j}({\bm{t}})\mathrm{diag}(Q_{1},\ldots,Q_{b})$ denotes the matrix in which the submatrix with row and column indices $J_{r}$ is $Q_{r}$), we obtain $R^{*}_{j}({\bm{t}})$ satisfying (2.2).

Doing this for all $j$, we obtain $(R^{*}_{1}({\bm{t}}),\ldots,R^{*}_{m}({\bm{t}}))$ which satisfies both conditions of Lemma 2.2. ∎

Lemma 2.3.

For each $\varepsilon>0$, there exists some ${\bm{t}}^{*}\in\mathbb{R}^{N}$ such that

$$\Big|\frac{d}{dt_{js}}f(\bm{t}^{*},R^{*}_{1}({\bm{t}}^{*}),\ldots,R^{*}_{m}({\bm{t}}^{*}))\Big|<\varepsilon$$

for all $t^{*}_{js}$.

Lemma 2.3 follows immediately from the following lemma which is introduced in [12, Lemma 3.4].

Lemma 2.4.

Let $A\subset{\mathbb{R}}^{d}$ be a compact set. Let $F:{\mathbb{R}}^{m}\times A\to\mathbb{R}$ and $y^{*}:{\mathbb{R}}^{m}\to A$ be functions satisfying the following properties:

1. $F(\bm{x},y)$ is bounded above and continuous on ${\mathbb{R}}^{m}\times A$.

2. For every $\bm{x}\in{\mathbb{R}}^{m}$, $F(\bm{x},y^{*}(\bm{x}))=\max_{y\in A}F(\bm{x},y)$.

3. For every $y\in A$, $F(\bm{x},y)$ as a function of $\bm{x}$ is differentiable on ${\mathbb{R}}^{m}$.

Then for each $\varepsilon>0$, there exists an $\bm{x}^{*}\in\mathbb{R}^{m}$ such that

$$\Big|\frac{d}{dx_{i}}F(\bm{x}^{*},y^{*}(\bm{x}^{*})\Big|<\varepsilon$$

for every $i\in[m]$.

Finally, we prove the following lemma.

Lemma 2.5.

Choose ${\bm{t}}^{*}\in\mathbb{R}^{N}$ and $(R^{*}_{1}({\bm{t}}^{*}),\ldots,R^{*}_{m}({\bm{t}}^{*}))$ that satisfy both conditions of Lemma 2.2 and Lemma 2.3 in the case of $\varepsilon=\frac{\varepsilon\cdot\min_{j\in[m]}p_{j}}{N}$. Then for each $j\in[m]$ and $s,s^{\prime}\in[n_{j}]$, $s\neq s^{\prime}$,

$$\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\langle{M_{i}{\bm{x}}_{a,js}},{M_{i}{\bm{x}}_{a,js^{\prime}}}\rangle=0$$

where $[{\bm{x}}_{a,j1},\ldots,{\bm{x}}_{a,jn_{j}}]=V_{a}^{T}R^{*}_{j}({\bm{t}}^{*})$.

Proof.

Fix $j_{0}\in[m]$. By Lemma 2.2, it is enough to consider the case $s\neq s^{\prime}$, $t_{js}\neq t_{js^{\prime}}$. Choose such $s$, $s^{\prime}$ and denote $s_{0}<s^{\prime}_{0}$.

Let $h:\mathbb{R}\to\mathbb{R}$ be the function

$$h(\theta)=\langle\bm{\gamma},{\bm{t}}\rangle-\sum^{k}_{i=1}\log\det\Big(\sum_{j\in[m]}p_{j}\sum_{a\in{\mathbf{A}}_{ij}}\sum_{s\in[n_{j}]}\exp{t_{js}}\cdot{\bm{x}}^{\prime}_{a,js}{\bm{x}}^{\prime T}_{a,js}\Big)$$

where

$$[{\bm{x}}^{\prime}_{a,i1},\ldots,{\bm{x}}^{\prime}_{a,jn_{j}}]=[{\bm{x}}_{a,j1},\ldots,{\bm{x}}_{a,jn_{j}}]R_{j}(\theta),\quad R_{j}(\theta)=I_{n_{j}}\quad(j\neq j_{0}),$$

$$R_{j_{0}}(\theta)=\begin{pmatrix}1\\ &\ddots\\ &&\cos{\theta}&\dots&\sin{\theta}\\ &&\vdots&&\vdots\\ &&-\sin{\theta}&\dots&\cos{\theta}\\ &&&&&\ddots\\ &&&&&&1\end{pmatrix}$$

For clarification, $R_{i_{0}}\in O_{n_{i_{0}}}(\mathbb{R})$ is obtained from the identity matrix by changing the $(s_{0},s_{0})$, $(s_{0}^{\prime},s_{0}^{\prime})$ entries to $\cos{\theta}$, the $(s_{0},s_{0}^{\prime})$ entry to $\sin{\theta}$, and the $(s_{0}^{\prime},s_{0})$ entry to $-\sin{\theta}$. Then

$$h(\theta) =f(\bm{t},R^{*}_{1}({\bm{t}}^{*}),..,R^{*}_{j_{0}}({\bm{t}}^{*})R_{j_{0}}(\theta),..,R^{*}_{m}({\bm{t}}^{*}))$$

$$\leq f(\bm{t},R^{*}_{1}({\bm{t}}^{*}),..,R^{*}_{j_{0}}({\bm{t}}^{*}),..,R^{*}_{m}({\bm{t}}^{*}))$$

$$=h(0)$$

for all $\theta\in\mathbb{R}$. Thus, $h(\theta)$ has the maximum at $\theta=0$. Using $\frac{d}{ds}\log\det A=\operatorname{tr}(A^{-1}\frac{d}{ds}A)$ for the invertible matrix $A$ (see, for example, [15, Chapter 9 ,Theorem 4]) we can calculate as follows:

$$0 =\frac{dh}{d\theta}(0)$$

$$=-\sum^{k}_{i=1}\operatorname{tr}\Bigg[X_{i}^{-1}\sum_{a\in{\mathbf{A}}_{ij_{0}}}\Big(p_{j_{0}}\exp{t^{*}_{j_{0}s_{0}^{\prime}}}\left.\frac{d}{d\theta}\right|_{\theta=0}{\bm{x}}^{\prime}_{a,j_{0}s_{0}}{\bm{x}}_{a,j_{0}s_{0}}^{\prime T}$$

$$\qquad\qquad\quad\qquad\qquad\quad\qquad\qquad\quad+p_{j_{0}}\exp{t^{*}_{j_{0}s_{0}^{\prime}}}\left.\frac{d}{d\theta}\right|_{\theta=0}{\bm{x}}^{\prime}_{a,j_{0}s_{0}^{\prime}}{\bm{x}}_{a,j_{0}s_{0}^{\prime}}^{\prime T}\Big)\Bigg]$$

$$=-p_{j_{0}}\exp{t^{*}_{j_{0}s_{0}}}\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij_{0}}}\operatorname{tr}\Bigg[\left.\frac{d}{d\theta}\right|_{\theta=0}(\cos\theta\,M_{i}{\bm{x}}_{a,j_{0}s_{0}}-\sin\theta\,M_{i}{\bm{x}}_{a,j_{0}s_{0}^{\prime}})$$

$$\qquad\qquad\quad\qquad\quad\qquad\qquad\quad\quad\qquad\quad\qquad\cdot(\cos\theta\,M_{i}{\bm{x}}_{a,j_{0}s_{0}}-\sin\theta\,M_{i}{\bm{x}}_{a,j_{0}s_{0}^{\prime}})^{T}\Bigg]$$

$$\qquad-p_{j_{0}}\exp{t^{*}_{j_{0}s_{0}^{\prime}}}\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij_{0}}}\operatorname{tr}\Bigg[\left.\frac{d}{d\theta}\right|_{\theta=0}(\sin\theta\,M_{i}{\bm{x}}_{a,j_{0}s_{0}}+\cos\theta\,M_{i}{\bm{x}}_{a,j_{0}s_{0}^{\prime}})$$

$$\qquad\qquad\qquad\quad\qquad\qquad\quad\qquad\quad\qquad\qquad\cdot(\sin\theta\,M_{i}{\bm{x}}_{a,j_{0}s_{0}}+\cos\theta\,M_{i}{\bm{x}}_{a,j_{0}s_{0}^{\prime}})^{T}\Bigg]$$

$$=-p_{j_{0}}\exp{t^{*}_{j_{0}s_{0}}}\Big(-2\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij_{0}}}\langle M_{i}{\bm{x}}_{a,j_{0}s_{0}},M_{i}{\bm{x}}_{a,j_{0}s_{0}^{\prime}}\rangle\Big)$$

$$\qquad\qquad\quad\qquad\quad\qquad\quad\qquad-p_{j_{0}}\exp{t^{*}_{j_{0}s_{0}^{\prime}}}\Big(2\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij_{0}}}\langle M_{i}{\bm{x}}_{a,j_{0}s_{0}},M_{i}{\bm{x}}_{a,j_{0}s_{0}^{\prime}}\rangle\Big)$$

$$=2p_{j_{0}}(\exp{t^{*}_{j_{0}s_{0}}}-\exp{t^{*}_{j_{0}s_{0}^{\prime}}})\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij_{0}}}\langle M_{i}{\bm{x}}_{a,j_{0}s_{0}},M_{i}{\bm{x}}_{a,j_{0}s_{0}^{\prime}}\rangle$$

Since $p_{j_{0}}>0$ and $t_{j_{0}s_{0}}\neq t_{j_{0}s_{0}^{\prime}}$, we have $\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij_{0}}}\langle{M_{i}{\bm{x}}_{a,js_{0}}},{M_{i}{\bm{x}}_{a,js_{0}^{\prime}}}\rangle=0$.

∎

2.3. Proof of Theorem 1.10

Using the previous lemmas, we construct a quiver datum that satisfies Theorem 1.10.

Proof of Theorem 1.10.

Choose ${\bm{t}}^{*}\in\mathbb{R}^{m}$ and $(R^{*}_{1}({\bm{t}}^{*}),\ldots,R^{*}_{m}({\bm{t}}^{*}))$ that satisfy Lemmas 2.2 and 2.3 in the case of $\varepsilon=\frac{\varepsilon\cdot\min_{j\in[m]}p_{j}}{N}$. Set

$$\mathbf{V}^{\prime}=(V_{a}^{\prime})_{a},\quad V_{a}^{\prime}=\begin{pmatrix}\exp{(\frac{t^{*}_{j1}}{2})}\ &&\text{\huge{0}}\\ &\ddots&\\ \text{\huge{0}}&&\exp{(\frac{t^{*}_{jn_{j}}}{2})}\end{pmatrix}R^{*}_{j}({\bm{t}}^{*})^{T}V_{a}M_{i}^{T}.$$

By the construction of $M_{i}$ and $R^{*}_{j}({\bm{t}}^{*})^{T}$, we can easily see the equivalence between $\mathbf{V}$ and $\mathbf{V}^{\prime}$. Also, for each $i\in[k]$,

$$\sum^{m}_{j=1}p_{j}\sum_{a\in{\mathbf{A}}_{ij}}(V_{a}^{\prime})^{T}V_{a}^{\prime}=M_{i}X_{i}M_{i}^{T}=id_{{\mathbb{R}}^{d_{i}}},$$

so $\mathbf{V}^{\prime}$ satisfies (1.9).

Finally, we prove (1.10). For $i\in[n],j\in[n_{i}]$, we define

$$\varepsilon_{js}=\frac{d}{dt_{js}}f(\bm{t}^{*},R^{*}_{1}({\bm{t}}^{*}),\ldots,R^{*}_{m}({\bm{t}}^{*}))\in\big(-\frac{\varepsilon\cdot\min_{j\in[m]}p_{j}}{N},\frac{\varepsilon\cdot\min_{j\in[m]}p_{j}}{N}\big).$$

Then by $\frac{d}{ds}\log{\mathrm{det(A)}}=\operatorname{tr}(A^{-1}\frac{d}{ds}A)$ for invertible matrix $A$,

$$\varepsilon_{js} =p_{j}-\sum^{k}_{i=1}\operatorname{tr}\Big(X_{i}^{-1}\sum_{a\in{\mathbf{A}}_{ij}}p_{j}\exp{t^{*}_{js}}\cdot\bm{x}_{a,js}{\bm{x}_{a,js}}^{T}\Big)$$

$$=p_{j}-p_{j}\exp{t^{*}_{js}}\cdot\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\operatorname{tr}(M_{i}\bm{x}_{a,js}{\bm{x}_{a,js}}^{T}M_{i}^{T})$$

$$=p_{j}-p_{j}\exp{t^{*}_{js}}\cdot\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\|M_{i}\bm{x}_{a,js}\|^{2}.$$

Thus

(2.3) $$\quad\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\|M_{i}\bm{x}_{a,js}\|^{2}=(1-\varepsilon_{js}{p_{j}}^{-1})\exp{(-t^{*}_{js})}.$$

Now,

$$\begin{pmatrix}\exp{(-\frac{t^{*}_{j1}}{2})}\ &\text{\huge{0}}\\ \quad\quad\quad\ \ddots&\\ \text{\huge{0}}&\exp{(-\frac{t^{*}_{jn_{j}}}{2})}\end{pmatrix}\Big(\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}V_{a}^{\prime}(V_{a}^{\prime})^{T}\Big)\begin{pmatrix}\exp{(-\frac{t^{*}_{j1}}{2})}\ &\text{\huge{0}}\\ \quad\quad\quad\ \ddots&\\ \text{\huge{0}}&\exp{(-\frac{t^{*}_{jn_{j}}}{2})}\end{pmatrix}$$

$$\qquad\qquad\qquad=\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\big(M_{i}V_{a}^{T}R^{*}_{j}(\bm{t}^{*})\big)^{T}\big(M_{i}V_{a}^{T}R^{*}_{j}(\bm{t}^{*})\big)$$

and using Lemma 2.5 and (2.3),

(1) $$\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\big( M_{i}V_{a}^{T}R^{*}_{j}(\bm{t}^{*})\big)^{T}\big(M_{i}V_{a}^{T}R^{*}_{j}(\bm{t}^{*})\big)$$

$$=\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}{\begin{pmatrix}(M_{i}{\bm{x}}_{a,j1})^{T}\\ :\\ :\\ (M_{i}{\bm{x}}_{a,jn_{j}})^{T}\end{pmatrix}}\Big[M_{i}{\bm{x}}_{a,j1},\ldots,M_{i}{\bm{x}}_{a,jn_{j}}\Big]$$

$$=\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\big(\langle M_{i}\bm{x}_{a,js},M_{i}{\bm{x}_{a,js^{\prime}}}\rangle\big)_{ss^{\prime}}$$

$$=\Big(\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\langle M_{i}\bm{x}_{a,js},M_{i}{\bm{x}_{a,js^{\prime}}}\rangle\Big)_{ss^{\prime}}$$

$$={\begin{pmatrix}(1-\varepsilon_{j1}{p_{j}}^{-1})\exp{(-t^{*}_{j1})}&\text{\huge{0}}\\ \qquad\qquad\qquad\quad\quad\ \ddots&\\ \text{\huge{0}}&(1-\varepsilon_{jn_{j}}{p_{j}}^{-1})\exp{(-t^{*}_{jn_{j}})}\end{pmatrix}}$$

for each $j\in[m]$. Therefore,

$$\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}V_{a}^{\prime}(V_{a}^{\prime})^{T}-id_{{\mathbb{R}}^{n_{j}}}=\begin{pmatrix}-\varepsilon_{j1}{p_{j}}^{-1}\ &\text{\huge{0}}\\ \quad\quad\quad\ \ddots&\\ \text{\huge{0}}&-\varepsilon_{jn_{j}}{p_{j}}^{-1}\end{pmatrix}.$$

Since $p_{j}>0$ for each $j\in[m]$, we see that $\mathbf{V}^{\prime}$ satisfies (1.10). ∎

3. Applications and Remarks

3.1. Characteristic of extremisable quiver data

Using the calculation in the proof of Theorem 1.10, we can prove that (1.7) and (1.8) are necessary for the existence of the extremiser (Theorem 1.6) introduced in [5] and [10].

Proof of the existence of the extremisers in Theorem 1.6.

For each $j\in[m]$, there exists a unique representation of $Y_{j}$ such that

$$Y_{j}:=R_{j}\begin{pmatrix}\exp{{t_{j1}}}\ &&\text{\huge{0}}\\ &\ddots&\\ \text{\huge{0}}&&\exp{t_{jn_{j}}}\end{pmatrix}(R_{j})^{T}$$

where $R_{j}\in O_{n_{j}}(\mathbb{R})$ and $\bm{t}_{j}=(t_{j1},\ldots,t_{jn_{j}})\in\mathbb{R}^{n_{j}}$. By the feasibility of $(\mathbf{V},\mathbf{p})$, $X_{i}$ is invertible. Also, since the tuple $((\bm{t}_{j})_{j\in[m]},R_{1},\ldots,R_{m})$ is the maximiser of the function $f$ defined in Section 2,

$$\frac{d}{dt_{js}}f((\bm{t}_{j})_{j\in[m]},R_{1},\ldots,R_{m})=0$$

for all $j\in[m]$, $s\in[n_{j}]$. Thus by rearranging the calculation (1), we obtain

(2) $$\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}V_{a}X_{i}^{-1}V_{a}^{T} =R_{j}R_{j}^{T}\Big(\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}V_{a}M_{i}^{T}M_{i}V_{a}^{T}\Big)R_{j}R_{j}^{T}$$

$$=R_{j}\Big(\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}\big(M_{i}V_{a}^{T}R_{j}\big)^{T}\big(M_{i}V_{a}^{T}R_{j}\big)\Big)R_{j}^{T}$$

$$=R_{j}\begin{pmatrix}\exp{(-{t_{j1}})}\ &\text{\huge{0}}\\ \qquad\quad\quad\ \ddots&\\ \text{\huge{0}}&\exp{(-{t_{jn_{j}}})}\end{pmatrix}R_{j}^{T}=Y_{j}^{-1}$$

for all $j\in[m]$.

∎

Remark. Choose any $\varepsilon>0$ from the proof of Theorem 1.10 again. Set

$$Y_{\varepsilon,j}:=R^{*}_{j}({\bm{t}}^{*})\begin{pmatrix}\exp{{t^{*}_{j1}}}\ &&\text{\huge{0}}\\ &\ddots&\\ \text{\huge{0}}&&\exp{t^{*}_{jn_{j}}}\end{pmatrix}R^{*}_{j}({\bm{t}}^{*})^{T},\quad\text{$j\in[m]$}.$$

By the above calculation (2), Lemma 2.3 is interpreted as $(Y_{\varepsilon,j})_{j}$ approaching some extremiser in a sense.

3.2. Ubiquity for the capacity

Using the result of Theorem 1.10, we can prove Theorem 1.9 in a different way from the proof introduced in [8] and [11]. The proof relies on the continuity of the capacity when $\mathbf{p}$ is a tuple of positive real numbers. (See [9].)

Proof of Theorem 1.9.

Set

$$\mathbf{F}(\mathbf{p}):=\{\mathbf{V}=(V_{a})_{a}:\text{$\mathrm{Cap}(\mathbf{V},\mathbf{p})>0$ and $\sum_{j\in[m]}p_{j}\sum_{a\in{\mathbf{A}}_{ij}}V^{T}_{a}V_{a}=id_{\mathbb{R}^{d_{i}}}$ for all $i\in[k]$}\}.$$

By the continuity of $\mathbf{V}\mapsto\mathrm{Cap}(\mathbf{V},\mathbf{p})$ (see [9, Theorem 3.6]), $\mathbf{F}(\mathbf{p})$ is closed. Since every $p_{j}$ is positive, $\mathbf{F}(\mathbf{p})$ is compact.

Let $\mathbf{V}^{(k)}$ be a tuple of matrices that satisfies the condition of Theorem 1.10 in the case of $\varepsilon=1/k$ for each $k\in\mathbb{N}$. By Proposition 1.5 and (1.9), $(\mathbf{V}^{(k)})_{k\in\mathbb{N}}\subset\mathbf{F}(\mathbf{p})$. Thus, the compactness of $\mathbf{F}(\mathbf{p})$ gives us a subsequence of $(\mathbf{V}^{(k)})_{k\in\mathbb{N}}$ and a tuple of matrices $\mathbf{G}\in\mathbf{F}(\mathbf{p})$ such that

$$\lim_{k\to\infty}\|\mathbf{G}-\mathbf{V}^{(k)}\|=0.$$

Furthermore, (1.10) shows us that

$$\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}G_{a}(G_{a})^{T}=\lim_{k\to\infty}\sum^{k}_{i=1}\sum_{a\in{\mathbf{A}}_{ij}}V_{a}^{(k)}(V_{a}^{(k)})^{T}=id_{{\mathbb{R}}^{n_{j}}}$$

for all $j\in[m]$. Therefore, $(\mathbf{G},\mathbf{p})$ is geometric.

∎

3.3. Upper bound on Brascamp–Lieb constants

From now on, we consider the case that $(\mathbf{B},\mathbf{p})$ is the Brascamp–Lieb datum with a tuple of linear maps $\mathbf{B}=(B_{j}\in\mathbb{R}^{n_{j}\times n})_{j\in[m]}$ and a tuple of positive exponents $\mathbf{p}=(p_{j})_{j\in[m]}$. In this section, we will provide an upper bound on the Brascamp–Lieb constant $\mathrm{BL}(\mathbf{B},\mathbf{p})$ using ubiquity.

First, we introduce $\mathbf{B}$-admissible sets. We denote by $\bm{1}_{H}$ the indicator vector of a set $H$. Following the paper of Dvir and Hu [12], a subset $H\subset[m]$ is called $\mathcal{V}$-admissible if the subspaces $\mathcal{V}=(V_{i}\subset\mathbb{R}^{l})_{i\in H}$ form a direct-sum decomposition of $\sum_{i=1}^{m}V_{i}$. Theorem 1.4 of [12] asserts that whenever

$$\textbf{p}\in\{\text{convex hull of $\bm{1}_{H}$: $H$ is $\mathcal{V}$-admissible}\},$$

the arrangement $(V_{i})$ can be transformed by an invertible linear map so that $\sum^{m}_{i=1}p_{i}\mathrm{Proj}_{V_{i}}$ is arbitrarily close to the identity operator.

Definition 3.1.

We say that a set $J\subset[m]$ is a $\mathbf{B}$-admissible basis set if

(3.1) $$\sum_{j\in J}\dim(\mathrm{Im}B_{j}^{T})=\dim\Big(\sum_{j\in J}\mathrm{Im}B_{j}^{T}\Big)=n.$$

Set $E:=\{(j,s):j\in[m],s\in[n_{j}]\}$. We say $H\subset E$ is a good basis set of $\mathbf{B}$ if there exists a $\mathbf{B}$-admissible basis set $J$ such that $H=\bigcup_{j\in J}\{(j,s):s\in[n_{j}]\}$.

Notation. We denote by $ab_{\mathbf{B}}$ the set of all $\mathbf{B}$-admissible basis sets and by $gb_{\mathbf{B}}$ the set of all good basis set of ${\mathbf{B}}$.

Based on the proof of [12, Lemma 3.1] (see also [2]), we have the following upper bound. Assume that $0=:0\log 0$.

Proposition 3.2.

Let $\mathbf{p}$ be in the convex hull of $(\bm{1}_{J})_{J\in ab_{\mathbf{B}}}$ and $\mathbf{p}=\sum_{J\in ab_{\mathbf{B}}}\lambda_{J}\bm{1}_{J}$ where $\sum_{J\in ab_{\mathbf{B}}}\lambda_{J}=1$ and $\lambda_{J}\geq 0$. Then we have

(3.2) $$\mathrm{BL}(\mathbf{B},\mathbf{p})\leq\exp{\frac{1}{2}\bigg(\sum_{J\in ab_{\mathbf{B}}}\lambda_{J}\log\lambda_{J}-\sum^{m}_{j=1}n_{j}p_{j}\log{p_{j}}\bigg)}\cdot\prod_{J\in ab_{\mathbf{B}}}\big|\mathrm{det}\big(({B_{j}^{T}})_{j\in J}\big)\big|^{-\lambda_{J}}.$$

In particular, we see that $(\mathbf{B},\mathbf{p})$ is feasible.

Proof.

See Section 2.1 again. We use $\mathcal{F}$ to denote the family of all $n$-subsets of $[N]$ and set $t_{H}=\sum_{e\in H}\exp{(t_{e}+\log\gamma_{e})}$ for each $H\in\mathcal{F}$. Based on the calculation in [12, Lemma 3.1], we obtain

$$\mathrm{det}\Big( \sum_{j\in[m],s\in[n_{j}]}p_{j}\exp{t_{js}}\bm{x}_{js}\bm{x}_{js}^{T}\Big)=\mathrm{det}\Bigg([\bm{x}_{11},\ldots,\bm{x}_{mn_{m}}]\begin{pmatrix}p_{1}\exp{t_{11}}\bm{x}_{11}^{T}\\ :\\ :\\ p_{m}\exp{t_{mn_{m}}}\bm{x}_{mn_{m}}^{T}\end{pmatrix}\Bigg)$$

$$=\sum_{H\in\mathcal{F}}\exp{t_{H}}\cdot\mathrm{det}\big((\bm{x}_{e})_{e\in H}\big)\cdot\mathrm{det}\big((\bm{x}_{e})^{T}_{e\in H}\big)\qquad(\text{Cauchy--Binet formula})$$

$$\geq\sum_{H\in gb_{\mathbf{B}},\lambda_{H}>0}\lambda_{H}\bigg(\frac{\exp{t_{H}}\cdot\mathrm{det}\big((\bm{x}_{e})_{e\in H}\big)^{2}}{\lambda_{H}}\bigg)$$

$$\geq\prod_{H\in gb_{\mathbf{B}},\lambda_{H}>0}\bigg(\frac{\exp{t_{H}}\cdot\mathrm{det}\big((\bm{x}_{e})_{e\in H}\big)^{2}}{\lambda_{H}}\bigg)^{\lambda_{H}}\qquad\qquad(\text{AM-GM inequality})$$

$$=\exp{\bigg(\sum^{m}_{j=1}n_{j}p_{j}\log{p_{j}}\bigg)}\cdot\exp{\langle\bm{\gamma},\bm{t}\rangle}\cdot\prod_{H\in gb_{\mathbf{B}},\lambda_{H}>0}\bigg(\frac{\mathrm{det}\big((\bm{x}_{e})_{e\in H}\big)^{2}}{\lambda_{H}}\bigg)^{\lambda_{H}}$$

where $[\bm{x}_{j1},\ldots,\bm{x}_{j1}]=B_{j}^{T}R_{j}$. Meanwhile, for each good basis set $H$, there exists an associated $\mathbf{B}$-admissible basis set $J=\{j_{1},\ldots,j_{k}\}$ such that

(3) $$\mathrm{det}\big((\bm{x}_{e})_{e\in H}\big) =\mathrm{det}(B^{T}_{j_{1}}R_{j_{1}},\ldots,B^{T}_{j_{k}}R_{j_{k}})$$

$$=\mathrm{det}(B^{T}_{j_{1}},\ldots,B^{T}_{j_{k}})\cdot\mathrm{det}\begin{pmatrix}R_{j_{1}}\ &&\text{\huge{0}}\\ &\ddots&\\ \text{\huge{0}}&&R_{j_{k}}\end{pmatrix}$$

$$=\mathrm{det}\big(({B^{T}_{j}})_{j\in J}\big).$$

Set $\lambda_{J}=\lambda_{H}$ for each associated ${\mathbf{B}}$-admissible basis set $J$. Using (3) and Theorem 1.1, we have

$$\mathrm{BL}(\mathbf{B},\mathbf{p}) \leq\exp{\frac{1}{2}\bigg(\sum_{H\in gb_{\mathbf{B}}}\lambda_{H}\log\lambda_{H}-\sum^{m}_{j=1}n_{j}p_{j}\log{p_{j}}\bigg)}\cdot\prod_{H\in gb_{\mathbf{B}}}\big|\mathrm{det}\big((\bm{x}_{e})_{e\in H}\big)\big|^{-\lambda_{H}}$$

$$=\exp{\frac{1}{2}\bigg(\sum_{J\in ab_{\mathbf{B}}}\lambda_{J}\log\lambda_{J}-\sum^{m}_{j=1}n_{j}p_{j}\log{p_{j}}\bigg)}\cdot\prod_{J\in ab_{\mathbf{B}}}\big|\mathrm{det}\big(({B^{T}_{j}})_{j\in J}\big)\big|^{-\lambda_{J}}.$$

∎

Next, using ubiquity, we provide an upper bound that gives an improvement for certain data. Assume that $\mathrm{BL}(\mathbf{B},\mathbf{p})<\infty$. Choose $(\bm{t}^{*},(R^{*}_{j}(\bm{t}^{*}))_{j\in[m]})$ in Lemma 2.5 and set

$$B_{j}^{\prime}=\begin{pmatrix}\|M{\bm{x}}_{j1}\|^{-1}\ &&\text{\huge{0}}\\ &\ddots&\\ \text{\huge{0}}&&\|M{\bm{x}}_{jn_{j}}\|^{-1}\end{pmatrix}(R^{*}_{j}(\bm{t}^{*}))^{T}B_{j}M^{T}$$

for every $j\in[m]$. Then $\mathbf{B}^{\prime}=(B_{j}^{\prime})_{j\in[m]}$ is equivalent to $\mathbf{B}$. Using arguments as in the proof of Theorem 1.9 and Theorem 1.10, observe that $\mathbf{B}^{\prime}$ satisfies the following properties.

Theorem 3.3.

We have

(3.3) $$B_{j}^{\prime}(B_{j}^{\prime})^{T}=id_{{\mathbb{R}}^{n_{j}}}\quad\text{for each $j\in[m]$ }$$

and

(3.4) $$\Big\|\sum^{m}_{j=1}p_{j}(B_{j}^{\prime})^{T}B_{j}^{\prime}-id_{{\mathbb{R}}^{n}}\Big\|<\varepsilon.$$

Theorem 3.4.

For any $k\in\mathbb{N}$, there exists an invertible matrix $M^{(k)}$ and a tuple of orthogonal matrices $(R^{(k)}_{j})_{j\in[m]}$ such that $\mathbf{B}^{(k)}=(B_{j}^{(k)})_{j\in[m]}$ satisfies

(3.5) $$B_{j}^{(k)}=\begin{pmatrix}\|M^{(k)}{\bm{x}}^{(k)}_{j1}\|^{-1}\ &&\emph{\huge{0}}\\ &\ddots\\ \emph{\huge{0}}&&\|M^{(k)}{\bm{x}}^{(k)}_{jn_{j}}\|^{-1}\end{pmatrix}(R_{j}^{(k)})^{T}B_{j}(M^{(k)})^{T}$$

and

(3.6) $$\|\mathbf{B}^{(k)}-\mathbf{G}\|<\frac{1}{k}\quad\text{for all $k\in\mathbb{N}$}$$

for some geometric datum $(\mathbf{G},\mathbf{p})$.

By the compactness of $\prod^{m}_{j=1}O_{n_{j}}(\mathbb{R})$, we obtain a subsequence of $k\in\mathbb{N}$ such that $R^{(k)}_{j}$ converges an orthogonal matrix $R^{\infty}_{j}$. Now, using (1.4), we have

(3.7) $$\mathrm{BL}(\mathbf{B},\mathbf{p})=\frac{\mathrm{det}(M^{{(k)}})}{\prod^{m}_{j=1}\prod^{n_{j}}_{s=1}\|M^{{(k)}}\bm{x}^{{(k)}}_{js}\|^{p_{j}}}\mathrm{BL}(\mathbf{B^{{(k)}}},\mathbf{p}).$$

Thus, by the continuity of $\mathbf{B}\mapsto\mathrm{BL}(\mathbf{B},\mathbf{p})$ [3, Theorem 1.1], $\mathrm{BL}(\mathbf{G},\mathbf{p})=1$ and (3.6), we have

$$\mathrm{BL}(\mathbf{B},\mathbf{p}) =\limsup_{k\to\infty}\frac{\mathrm{det}(M^{{(k)}})}{\prod^{m}_{j=1}\prod^{n_{j}}_{s=1}\|M^{{(k)}}\bm{x}^{{(k)}}_{js}\|^{p_{j}}}\mathrm{BL}(\mathbf{B^{{(k)}}},\mathbf{p})$$

$$\leq\limsup_{k\to\infty}\frac{\mathrm{det}(M^{{(k)}})}{\prod^{m}_{j=1}\prod^{n_{j}}_{s=1}\|M^{{(k)}}\bm{x}^{{(k)}}_{js}\|^{p_{j}}}.$$

Definition 3.5.

Set $(\bm{g}_{j1},\ldots,\bm{g}_{jn_{j}})=B_{j}^{T}R^{\infty}_{j}$. We say that a set $B\subset E$ is a limit basis set if $(\bm{g}_{e})_{e\in B}$ is a basis of $\mathbb{R}^{n}$.

Notation. We denote by $b_{\infty}$ the set of all limit basis sets.

Remark. Immediately we see that if $\mathbf{p}=(p_{j})_{j}$ is in the convex hull of $(\bm{1}_{J})_{J\in ab_{\mathbf{B}}}$, $\bm{\gamma}=(p_{j})_{js}$ is in the convex hull of $(\bm{1}_{H})_{H\in gb_{\mathbf{B}}}$ and also in the convex hull of $(\bm{1}_{B})_{B\in b_{\infty}}$.

Lemma 3.6.

Let $(\mathbf{B},\mathbf{p})$ be a feasible datum. Let $\bm{\gamma}=(\gamma_{js})_{(j,s)\in E}$, $\gamma_{ji}=p_{j}$ be in the convex hull of $(\bm{1}_{B})_{B\in b_{\infty}}$ and $\bm{\gamma}=\sum_{B\in b_{\infty}}\lambda_{B}\bm{1}_{B}$ where $\sum_{B\in b_{\infty}}\lambda_{B}=1$ and $\lambda_{B}\geq 0$. Then there exists a tuple of non-negative reals $(\lambda_{B})_{B\in b_{\infty}}$ such that

(3.8) $$\mathrm{BL}(\mathbf{B},\mathbf{p})\leq\prod_{B\in b_{\infty}}\big|\mathrm{det}\big((\bm{g}_{e})_{e\in B}\big)\big|^{-\lambda_{B}},\quad\sum_{B\in b_{\infty}}\lambda_{B}=1.$$

Proof.

By the convergence of $(R^{(k)}_{j})_{j\in[m]}$ and the continuity of $A\mapsto\mathrm{det}(A)$, we see that for sufficiently large $k\in\mathbb{N}$, $(\bm{x}^{(k)}_{e})_{e\in B}$ is the basis of $\mathbb{R}^{n}$ for every $B\in b_{\infty}$. Using Hadmard’s inequality, we have

(3.9) $$|\mathrm{det}(M^{(k)})|\big|\mathrm{det}\big((\bm{x}^{(k)}_{e})_{e\in B}\big)\big|=\big|\mathrm{det}\big((M^{(k)}\bm{x}^{(k)}_{e})_{e\in B}\big)\big|\leq\prod_{e\in B}\|M^{(k)}\bm{x}^{(k)}_{e}\|.$$

Hence, we have

$$\frac{\mathrm{det}(M^{{(k)}})}{\prod^{m}_{j=1}\prod^{n_{j}}_{s=1}\|M^{{(k)}}\bm{x}^{{(k)}}_{js}\|^{p_{j}}} =\frac{\mathrm{det}(M^{{(k)}})^{\sum_{B\in b_{\infty}}\lambda_{B}}}{\prod_{B\in b_{\infty}}\prod_{e\in B}\|M^{{(k)}}\bm{x}^{{(k)}}_{e}\|^{\lambda_{B}}}$$

$$=\prod_{B\in b_{\infty}}\Bigg(\frac{\mathrm{det}(M^{{(k)}})}{\prod_{e\in B}\|M^{{(k)}}\bm{x}^{{(k)}}_{e}\|}\Bigg)^{\lambda_{B}}$$

$$\leq\prod_{B\in b_{\infty}}\big|\mathrm{det}\big((\bm{x}^{(k)}_{e})_{e\in B}\big)\big|^{-\lambda_{B}}$$

for any sufficiently large $k\in\mathbb{N}$. Hence we obtain

$$\mathrm{BL}(\mathbf{B},\mathbf{p}) \leq\limsup_{k\to\infty}\frac{\mathrm{det}(M^{{(k)}})}{\prod^{m}_{j=1}\prod^{n_{j}}_{s=1}\|M^{{(k)}}\bm{x}^{{(k)}}_{js}\|^{p_{j}}}\leq\lim_{k\to\infty}\prod_{B\in b_{\infty}}\big|\mathrm{det}\big((\bm{x}^{(k)}_{e})_{e\in B}\big)\big|^{-\lambda_{B}}$$

$$=\prod_{B\in b_{\infty}}\big|\mathrm{det}\big((\bm{g}_{e})_{e\in B}\big)\big|^{-\lambda_{B}}.$$

∎

Observe that using Proposition 3.2, Lemma 3.6 and (3) for $R^{\infty}_{j}\in O_{n_{j}}(\mathbb{R})$, we have the following.

Theorem 3.7.

Let $\mathbf{p}$ be in the convex hull of $(\bm{1}_{J})_{J\in ab_{\mathbf{B}}}$ and $\mathbf{p}=\sum_{J\in ab_{\mathbf{B}}}\lambda_{J}\bm{1}_{J}$ where $\sum_{J\in ab_{\mathbf{B}}}\lambda_{J}=1$ and $\lambda_{J}\geq 0$. Then we have

(3.10) $$\mathrm{BL}(\mathbf{B},\mathbf{p})\leq\prod_{J\in ab_{\mathbf{B}}}\big|\mathrm{det}\big(({B^{T}_{j}})_{j\in J}\big)\big|^{-\lambda_{J}}.$$

Remark.

(1) In the rank-one case, Barthe has shown that the assumption of Theorem 3.7 is a necessary and sufficient condition for $(\mathbf{B},\mathbf{p})$ to be feasible [2].

(2) An argument based on multilinear interpolation can also be used to prove (3.10).

Theorem 3.7 gives us the following corollary.

Corollary 3.8.

Let $\mathbf{p}$ be in the convex hull of $(\bm{1}_{J})_{J\in ab_{\mathbf{B}}}$. Let $B_{j}$ be a matrix of which every entry is the form of $\frac{c_{kl}}{d}$ where $(c_{kl})_{kl}\in\mathbb{Z}^{n_{j}\times n}$ and $d\in\mathbb{N}$. Then we have

(3.11) $$\mathrm{BL}(\mathbf{B},\mathbf{p})\leq d^{n}.$$

Proof.

For each $\mathbf{B}$-admissible basis set $J$, we have

$$\mathrm{det}\big(({B^{T}_{j}})_{j\in J}\big)=d^{-n}\mathrm{det}(C_{J})$$

where $C_{J}$ is the matrix with integer entries. Since $\mathrm{det}(C_{J})\in\mathbb{Z}$, by Theorem 3.7, we obtain

$$\mathrm{BL}(\mathbf{B},\mathbf{p})\leq\prod_{J\in ab_{\mathbf{B}}}\big|d^{-n}\mathrm{det}(C_{J})\big|^{-\lambda_{J}}=d^{n}\cdot\prod_{J\in ab_{\mathbf{B}}}\big|\mathrm{det}(C_{J})\big|^{-\lambda_{J}}\leq d^{n}.$$

∎

Acknowledgments. The author would like to express sincere gratitude to her advisor, Neal Bez, for his continuous guidance, encouragement, and invaluable suggestions throughout the preparation of this paper. His insightful comments and careful reading of earlier drafts greatly improved the content of this work.

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

Email address: mireille-labeille@g.ecc.u-tokyo.ac.jp