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# On the topology of some hyperspaces of convex bodies associated to tensor
norms

Luisa F. Higueras-Montaño and Natalia Jonard-Pérez  Departamento de
Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México,
04510 Ciudad de México, México.  (L. Higueras-Montaño<EMAIL_ADDRESS>(N.
Jonard-Pérez<EMAIL_ADDRESS>
###### Abstract.

For every tuple $d_{1},\dots,d_{l}\geq 2,$ let
$\mathbb{R}^{d_{1}}\otimes\cdots\otimes\mathbb{R}^{d_{l}}$ denote the tensor
product of $\mathbb{R}^{d_{i}},$ $i=1,\dots,l.$ Let us denote by
$\mathcal{B}(d)$ the hyperspace of centrally symmetric convex bodies in
$\mathbb{R}^{d},$ $d=d_{1}\cdots d_{l},$ endowed with the Hausdorff distance,
and by $\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ the subset of
$\mathcal{B}(d)$ consisting of the convex bodies that are closed unit balls of
reasonable crossnorms on
$\mathbb{R}^{d_{1}}\otimes\cdots\otimes\mathbb{R}^{d_{l}}.$ It is known that
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ is a closed, contractible and
locally compact subset of $\mathcal{B}(d).$ The hyperspace
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ is called the space of tensorial
bodies. In this work we determine the homeomorphism type of
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l}).$ We show that even if
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ is not closed with respect to the
Minkowski sum, it is an absolute retract homeomorphic to
$\mathcal{Q}\times\mathbb{R}^{p},$ where $\mathcal{Q}$ is the Hilbert cube and
$p=\frac{d_{1}(d_{1}+1)+\cdots+d_{l}(d_{l}+1)}{2}.$ Among other results, the
relation between the Banach-Mazur compactum and the Banach-Mazur type
compactum associated to $\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ is
examined.

###### Key words and phrases:

Infinite dimensional topology, absolute neighborhood retract, convex body,
tensor norm, hyperspace, tensor product of convex sets, Banach-Mazur compactum

###### 2020 Mathematics Subject Classification:

57N20, 46M05, 52A21, 57S20, 54C55, 15A69.

The first author has been supported by The Post-Doctoral Scholarship Program
at UNAM. The first and second author have been supported by CONACyT grant
252849 (México) and by PAPIIT grant IN115819 (UNAM, México)

##  1\. Introduction

In the last 50 years a lot of research on the topological properties of some
hyperspaces of convex compacta in the Euclidean space $\mathbb{R}^{d}$ has
been carried out. Among others, the topological structure of the following
hyperspaces has been determined: the non-empty compact convex sets
$cc(\mathbb{R}^{d})$ [25]; the centrally symmetric convex bodies
$\mathcal{B}(d)$ [1, 2]; the class of convex bodies $\mathcal{K}_{d}$ [5, 23];
the convex bodies of constant width [6, 10, 11]; and, recently, the convex
bodies with smooth boundary [12]. As can be traced in the seminal work [25],
the fundamental tool to tackle these problems has been the theory of Hilbert
cube manifolds. A Hilbert cube manifold is a separable and metrizable space
modeled on the Hilbert cube (see Section 1.1). The Hilbert cube $\mathcal{Q}$
is the space $\prod_{i=1}^{\infty}[-1,1]$ endowed with the product topology.

Some of the previous hyperspaces arise naturally in contexts different from
Convex Geometry or Geometric Topology. This is the case of $\mathcal{B}(d),$
which emerges from the theory of Banach spaces. It is well-known that the
convex bodies in $\mathcal{B}(d)$ are, precisely, the unit balls of norms on
$\mathbb{R}^{d}$ (see for instance [32, Section 1.7] or [33, Section 1.1]). In
the same vein, a hyperspace arising from the theory of tensor norms on Banach
spaces was introduced in [18, 21]. It is called the space of tensorial bodies
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l}),$ $d_{i}\in\mathbb{N},$
$i=1,\dots,l,$ and it is composed by the convex bodies
$P\subset\mathbb{R}^{d},$ $d=d_{1}\cdots d_{l},$ that are unit balls of Banach
spaces, whose norms are reasonable crossnorms. The main goal of this paper is
to calculate the topological structure of this hyperspace. Our approach to the
problem is based on the theory of Hilbert cube manifolds and topological
groups (see Sections 1.1 and 1.2).

Before giving a formal definition of
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$, it is convenient to recall some
facts about the so called theory of tensor norms. We are interested in the
finite-dimensional setting. A norm $\alpha(\cdot)$ on the tensor product of
finite-dimensional Banach spaces
$\otimes_{i=1}^{l}(\mathbb{R}^{d_{i}},\|\cdot\|_{i})$ is called a reasonable
crossnorm if and only if

$\epsilon\left(u\right)\leq\alpha\left(u\right)\leq\pi\left(u\right)\text{ for
every }u\in\otimes_{i=1}^{l}(\mathbb{R}^{d_{i}},\|\cdot\|_{i}).$

The norms $\epsilon(\cdot)$ and $\pi(\cdot)$ are the injective and the
projective tensor norms, respectively. Actually, they are the smallest and the
biggest reasonable crossnorms (we refer to [16, 30] for the definitions and
basic properties). In [18, Theorem 3.2], a characterization of the convex
bodies in $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}\simeq\mathbb{R}^{d},$
$d=d_{1},\dots,d_{l},$ that are closed unit balls of reasonable crossnorms is
given. That is, an explicit description of the convex bodies
$P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ for which there exists norms
$\|\cdot\|_{i},$ $i=1,\dots,l,$ (not determined a priori) such that $P$ is the
closed unit ball of a reasonable crossnorm on
$\otimes_{i=1}^{l}(\mathbb{R}^{d_{i}},\|\cdot\|_{i})$ is established. With it,
the class of tensorial bodies in $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ is
introduced [18, Definition 3.3]. A centrally symmetric ($0$-symmetric) convex
body $P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ is called tensorial body if
there exists $0$-symmetric convex bodies $P_{i}\subset\mathbb{R}^{d_{i}},$
$i=1,\dots,l,$ such that

$P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\subseteq P\subseteq
P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}.$

Where, $\otimes_{\pi}$ and $\otimes_{\epsilon}$ are the projective and the
injective tensor products of convex bodies, respectively (see Section 2). The
hyperspace associated to the tensorial bodies
$\mathcal{B}_{\otimes}(d_{1},\dots.,d_{l})$ is introduced in [21]. There, some
of its fundamental properties are exhibited. It is known that
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ carries a natural structure of a
$GL_{\otimes}$-proper space. In this case, $GL_{\otimes}(d_{1},\dots,d_{l})$
is the group consisting of linear isomorphisms on
$\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ preserving decomposable vectors (see
Section 1.2). The action is given by the evaluation $TP,$ for every $T\in
GL_{\otimes}(d_{1},\dots,d_{l})$ and
$P\in\mathcal{B}_{\otimes}(d_{1},\dots,d_{l}).$ The associated orbit space
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$ is
a metrizable compact space called the compactum of tensorial bodies [21,
Corollary 4.9].

The main results of the paper are Theorem 4.3 and Corollary 5.5. In Theorem
4.3, we show that $\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ is an absolute
retract (see Section 1.1 for the definition). In Corollary 5.5, we calculate
the topological structure of the space of tensorial bodies. It is proved that

(1.1) $\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})\textit{ is homeomorphic to
}\mathcal{Q}\times\mathbb{R}^{p},\
p=\frac{d_{1}(d_{1}+1)+\cdots+d_{l}(d_{l}+1)}{2},$

for every tuple $d_{i}\geq 2,$ $i=1,\dots,l.$ It is known that in
$\mathbb{R}\otimes\mathbb{R}^{d}$ the tensor structure is trivial, and
$\mathcal{B}_{\otimes}(1,d)$ coincides with $\mathcal{B}(d)$ [18, Proposition
3.8].

We now briefly describe the contents of the paper. In Sections 1.1 and 1.2, we
introduce the notation and basic results on Hilbert cube manifolds and the
theory of $G$-spaces. Then, in Section 2, we review the fundamental properties
of the space of tensorial bodies [18, 21]. In Proposition 2.2 and Theorem 2.3,
we summarize the results from [21] that we use frequently in the work. In this
sense, it is recalled that

(1.2) $\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})\textit{ is homeomorphic to
}\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})\times\mathbb{R}^{p}$

where $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ is the $O_{\otimes}$-global
slice for the $GL_{\otimes}$-space $\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$
introduced in [21, Section 4.1], and $p$ is the same as (1.1). Here,
$O_{\otimes}(d_{1},\dots,d_{l})$ is the closed subgroup of
$GL_{\otimes}(d_{1},\dots,d_{l})$ consisting of orthogonal maps (Section 1.2).
Through the work, the letters $d,d_{i}\geq 2,$ $i=1,\dots,l,$ will denote
natural numbers. The proposition 2.4 is new. It shows that the Hilbertian
tensor product $\otimes_{2}$ of $0$-symmetric ellipsoids is continuous with
respect to the Hausdorff distance.

In Section 3, we completely describe the topological structure of the
hyperspace
$\Pi(d_{1},\dots,d_{l})=\\{P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}:P_{i}\in\mathcal{B}(d_{i})\\},$
determined by the projective tensor product $\otimes_{\pi}$. We show that the
$GL_{\otimes}$-map
$conv_{\otimes}:\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})\rightarrow\Pi(d_{1},\dots,d_{l}),$
defined in (2.6), is a $GL_{\otimes}$-deformation retraction onto
$\Pi(d_{1},\dots,d_{l}),$ and it is also a proper map (Proposition 3.1). The
main result of this section is Theorem 3.2. There, we prove that
$\Pi(d_{1},\dots,d_{l})$ is a Hilbert cube manifold homeomorphic to
$\mathcal{Q}\times\mathbb{R}^{p},$ with $p$ as in (1.1).

Determining whether or not a metrizable locally compact space is an absolute
neighborhood retract ANR (see Section 1.1 for the definition) is a fundamental
step in order to prove that it is a Hilbert cube manifold. Section 4 is
devoted to the study of the ANR property on
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$. First, in Proposition 4.1, we show
that $\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ is not closed under the
Minkowski sum. Establishing, in consequence, a notable difference with other
hyperspaces such as $\mathcal{K}_{d}$ and $\mathcal{B}(d).$ Despite this
failure, in Proposition 4.2, we give conditions under which the Minkowski sum
of tensorial bodies remains in the class. Finally, in Theorem 4.3, we prove
that $\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ is an absolute retract. This
is done by means of exhibiting a retraction from the space of $0$-symmetric
convex bodies in $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ onto the space of
tensorial bodies.

The topological structure of $\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ is
calculated in Section 5. Our approach consist in proving that
$\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ is homeomorphic to $\mathcal{Q}$
(see (1.2) above). To address this problem, we use Torunczyk’s
characterization of Hilbert cube manifolds [35, Theorem 1] and a classical
result of R. Edwards [35, §3]. In this way, in Proposition 5.3, we first
exhibit a $Z$-set in $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l}),$ then we prove
that the complement of such a set is an open Hilbert cube manifold in
$\mathcal{L}_{\otimes}(d_{1},\dots,d_{l}).$ This allows us to show that

$\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})\textit{ is homeomorphic to
}\mathcal{Q}.$

This result appears as Theorem 5.4. In Proposition 5.1, it is showed that the
hyperspace
$\Pi(d_{1},\dots,d_{l})\cap\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ is an
$O_{\otimes}$-deformation retraction of
$\mathcal{L}_{\otimes}(d_{1},\dots,d_{l}),$ and it is homeomorphic to
$\mathcal{Q}.$

We finish the paper by exploring the topological structure of the compactum of
tensorial bodies
$\mathcal{B}_{\otimes}(d_{1},\dots.,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l}).$
Among others, we show that it is a contractible space (Proposition 6.1). Some
remarks and open questions on the relation between the compactum
$\mathcal{B}_{\otimes}(d_{1},\dots.,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$
and the so called Banach-Mazur compactum $\mathcal{BM}(d)$ are detailed. In
Proposition 6.3, we show that the orbit space
$\Pi(d_{1},\dots,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$ is homeomorphic to
$\mathcal{BM}(d_{1})\times\cdots\times\mathcal{BM}(d_{l}),$ as long as the
dimensions $d_{i},$ $i=1,\dots,l,$ are different from each other.

###  1.1. Notation and basic definitions

Let $X$ be a topological space and let $A\subseteq X$ be closed. We say that
$A$ is a retract of $X$ if there exists a continuous surjective map
$r:X\rightarrow A$ such that $r(a)=a,$ for all $a\in A.$ Such a map $r$ is
called a retraction. A metrizable space $X$ is called an absolute neighborhood
retract (ANR) if for every metrizable space $Z$ containing $X$ as closed
subset, there exists a neighborhood $U$ of $X$ in $Z$ and a retraction
$r:U\rightarrow X.$ In this case, we write $X\in ANR.$ If we can always take
$U=Z,$ we say that $X$ is an absolute retract (AR), and we write $X\in AR.$ It
is well-known that every closed convex subset of a Banach space is an AR [24,
Theorem 1.5.1].

A continuous map $f:X\rightarrow Y$ between topological spaces is called
proper if for every compact subset $K\subseteq Y,$ $f^{-1}(K)$ is compact too.
A proper and surjective map $f:X\rightarrow Y$ between ANR’s is called cell-
like if every fiber $f^{-1}(y),$ $y\in Y,$ has the property $UV^{\infty}.$
That is, for each neighborhood $U$ of $f^{-1}(y)$ there exists a neighborhood
$V\subset U$ of $f^{-1}(y)$ such that the inclusion $V\hookrightarrow U$ is
homotopic to a constant map from $V$ to $U.$ In particular, if all of the
fibers $f^{-1}(y)$ are contractible, then $f$ has the property $UV^{\infty}$
(see [15, p. 91]). Recall that a topological space is called contractible if
there exists a homotopy $H:X\times[0,1]\rightarrow X$ such that $H(\cdot,0)$
is the identity on $X$ and $H(\cdot,1)$ is a constant map.

The Hilbert cube $\mathcal{Q}$ is the topological product
$\prod_{i=1}^{\infty}[-1,1].$ A Hilbert cube manifold (or
$\mathcal{Q}$-manifold) is a separable metrizable space that admits an open
cover by sets homeomorphic to open subsets of $\mathcal{Q}.$ A fundamental
result on Hilbert cube manifolds asserts that $\mathcal{Q}$ is the only
$\mathcal{Q}$-manifold which is compact and contractible [24, Theorem 7.5.8].
We refer to [15, 24, 35] for the fundamentals of the theory of
$\mathcal{Q}$-manifolds and infinite-dimensional topology.

A closed subset $A$ of a metric space $X$ is a $Z$-set if $\\{f\in
C(\mathcal{Q},X):f(\mathcal{Q})\cap A=\emptyset\\}$ is dense in
$C(\mathcal{Q},X).$ Here, $C(\mathcal{Q},X)$ is the space of continuous maps
from $\mathcal{Q}$ to $X$ endowed with the compact-open topology.

###  1.2. G-spaces

We recall some concepts from the theory of $G$-spaces that will be used
throughout the work. By a $G$-space we mean a Tychonoff space $X$ equipped
with a continuous action $G\times X\rightarrow X$ of a topological group $G$
on $X.$ The image under the action of the pair $(g,x)$ is denoted by $gx.$ If
$S$ is a subset of a $G$-space $X$ and $H\subseteq G$ is a subgroup, then
$H(S):=\\{hs:h\in H,s\in S\\}$ denotes the $H$-saturation of $S.$ In
particular, $G(x)$ denotes the orbit of $x\in X.$ As usual, by $X/G$ we denote
the orbit space. We say that a subset $S\subseteq X$ is $H$-invariant if
$H(S)=S.$

A continuous map $f:X\rightarrow Y$ between $G$-spaces is called equivariant
or simply a $G$-map if $f(gx)=gf(x),$ for any $g\in G$ and $x\in X.$ A $G$-map
$r:X\rightarrow A,$ $A\subseteq X,$ is called $G$-strong deformation
retraction, if there is a homotopy $H:X\times[0,1]\rightarrow X$ such that:
$H_{0}$ is the identity on $X;$ $H_{1}=r;$ $H(a,t)=a$ for each $t\in[0,1]$ and
$a\in A;$ and $H_{t}$ is a $G$-map for every $t.$ In this context, $A$ is
called $G$-strong deformation retract. For a deeper discussion of $G$-spaces,
we refer the reader to [14].

A G-space $X$ is called proper (in the sense of Palais [28]), if it has an
open cover such that every $x\in X$ has a neighborhood $V$ for which the set
$\\{g\in G:gS\cap V\neq\emptyset\\}$ has compact closure in $G$. We recall the
definition of a slice. Let $X$ be a $G$-space and $H$ a closed subgroup of
$G.$ An $H$-invariant subset $S\subseteq X$ is called an $H$-slice in $X,$ if
$G\left(S\right)$ is open in $X$ and there exists a $G$-map
$f:G\left(S\right)\rightarrow G/H$ such that $S=f^{-1}\left(eH\right).$ If
$G\left(S\right)=X,$ $S$ is called a global $H$-slice of $X$ (see e.g. [27, p.
305]).

In this work, we are interested in $G$-spaces for which $G$ is a closed
subgroup of the general lineal group $GL(d)$. Recall that $GL(d)$ consists of
the linear isomomorphisms $T:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}.$ The
topology on $GL(d)$ is the one induced by the operator norm
$\|T\|=\text{sup}_{x\in B_{2}^{d}}\|T(x)\|.$ Here $B_{2}^{d}$ denotes the
Euclidean ball associated to the standard Euclidean norm $\|\cdot\|$ on
$\mathbb{R}^{d}.$ As usual, by $\mathbb{S}^{d-1}$ and
$\langle\cdot,\cdot\rangle,$ we denote the sphere $\partial B_{2}^{d}$ and the
standard scalar product on $\mathbb{R}^{d},$ respectively.

Since we will deal with groups associated to tensor products of vector spaces,
we briefly recall the following. The tensor product
$\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ can be endowed with a Euclidean
structure. This is the one determined by the Hilbert tensor product of
Euclidean spaces, and it is induced by the scalar product
$\langle\cdot,\cdot\rangle_{H},$

(1.3) $\left\langle x^{1}\otimes\cdots\otimes x^{l},y^{1}\otimes\cdots\otimes
y^{l}\right\rangle_{H}:={\prod}_{i=1}^{l}\left\langle
x^{i},y^{i}\right\rangle.$

We simply write the definition of $\langle\cdot,\cdot\rangle_{H}$ on
decomposable vectors $x^{1}\otimes\cdots\otimes x^{l}$ and
$y^{1}\otimes\cdots\otimes y^{l}$ of $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}.$
However, it can be extended to the whole space by multilinearity [22, Section
2.5]. The space $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ together with the scalar
product $\langle\cdot,\cdot\rangle_{H}$ is denoted by
$\otimes_{H,i=1}^{l}\mathbb{R}^{d_{i}}.$ Its corresponding norm and Euclidean
ball are denoted by $\|\cdot\|_{H}$ and $B_{2}^{d_{1},\dots,d_{l}},$
respectively. The set of decomposable vectors in
$\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ is denoted by
$\Sigma_{d_{1},\ldots,d_{l}}.$ That is,

$\Sigma_{d_{1},\ldots,d_{l}}:=\\{x^{1}\otimes\cdots\otimes
x^{l}:x^{i}\in\mathbb{R}^{d_{i}},i=1,\ldots,l\\}.$

By $GL_{\otimes}(d_{1},\ldots,d_{l}),$ we denote the group of linear
isomorphisms on $\otimes_{H,i=1}^{l}\mathbb{R}^{d_{i}}$ sending decomposable
vectors into decomposable vectors. $O_{\otimes}(d_{1},\ldots,d_{l})$ denotes
the subgroup of $GL_{\otimes}(d_{1},\ldots,d_{l})$ consisting of orthogonal
maps. In [21, Section 4], it is proved that both
$GL_{\otimes}(d_{1},\ldots,d_{l})$ and $O_{\otimes}(d_{1},\ldots,d_{l})$ are
closed subgroups of $GL(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}).$ In particular,
they are Lie groups and $O_{\otimes}(d_{1},\ldots,d_{l})$ is a maximal compact
subgroup of $GL_{\otimes}(d_{1},\ldots,d_{l})$ [21, Appendix A]. To shorten
notation, we simply write $GL_{\otimes}$ and $O_{\otimes}.$

Given $T\in GL_{\otimes}(d_{1},\ldots,d_{l}),$ there exists $T_{i}\in
GL(d_{i}),$ $i=1,\ldots,l,$ and a permutation $\sigma$ on $\\{1,\ldots,l\\}$
such that $T=(T_{1}\otimes\cdots\otimes T_{l})U_{\sigma}.$ Here,
$U_{\sigma}\in GL_{\otimes}$ is defined in decomposable vectors as

$U_{\sigma}(x^{1}\otimes\cdots\otimes x^{l})=x^{\sigma(1)}\otimes\cdots\otimes
x^{\sigma(l)},$

and $\sigma$ is such that $x^{\sigma(1)}\otimes\cdots\otimes
x^{\sigma(l)}\in\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ for
$x^{i}\in\mathbb{R}^{d_{i}},$ $i=1,\ldots,l.$ The subgroup of $GL_{\otimes}$
consisting of the maps $U_{\sigma}$ defined above is denoted by $\mathcal{P}.$
It is worth noticing that $\mathcal{P}\subset O_{\otimes}$. Indeed, the
standard basis $e_{k_{i}}^{d_{i}},$ $k_{i}=1,\ldots,d_{i},$ of
$\mathbb{R}^{d_{i}}$ is such that
$U_{\sigma}(e_{k_{1}}^{d_{1}}\otimes\cdots\otimes
e_{k_{l}}^{d_{l}})=e_{k_{\sigma(1)}}^{d_{\sigma(1)}}\otimes\cdots\otimes
e_{k_{\sigma(l)}}^{d_{\sigma(l)}}.$ For a complete description of the
properties of $\mathcal{P}$ and its relation with $GL_{\otimes},$ we refer to
[21, Lemma A.1 and Proposition A.4].

##  2\. Preliminaries on tensorial bodies

We first introduce some results from convex geometry that we will be used in
the work. The letters $d,d_{i}\geq 2,$ with $i=1,\dots,l,$ will always denote
natural numbers. A convex compact subset $P\subset\mathbb{R}^{d}$ with non-
empty interior is called a convex body. If in addition $P=-P,$ we say that $P$
is a $0$-symmetric convex body. By $\mathcal{K}_{d}$ and $\mathcal{B}(d),$ we
denote, respectively, the classes of convex bodies and $0$-symmetric convex
bodies in $\mathbb{R}^{d}.$ The polar body of $P\in\mathcal{B}(d)$ is defined
as $P^{\circ}=\\{y\in\mathbb{R}^{d}:\langle x,y\rangle\leq 1\text{ for all
}x\in P\\}.$ It is well-known that $P^{\circ}\in\mathcal{B}(d).$ The Minkowski
functional associated to $P\in\mathcal{K}_{d}$ is defined by
$g_{P}(x)=\text{inf}\\{\lambda>0:\lambda^{-1}x\in P\\},$ for every
$x\in\mathbb{R}^{d}.$ In the case of $0$-symmetric convex bodies, the
Minkowski functional provides a correspondence between norms on
$\mathbb{R}^{d}$ and $0$-symmetric convex bodies. Indeed, the map sending
$P\in\mathcal{B}(d)$ to its Minkowski functional $g_{P}(\cdot)$ is a
bijection; the unit ball of $(\mathbb{R}^{d},g_{P})$ is $P;$ and
$g_{P^{\circ}}(x)=\left\|\langle\cdot,x\rangle\right\|,$ where
$\langle\cdot,x\rangle$ is regarded as a linear functional from
$\left(\mathbb{R}^{d},g_{P}\right)$ to $\mathbb{R}.$ See [32, Remark 1.7.8].
The Hausdorff distance between non-empty compact sets
$P,R\subset\mathbb{R}^{d}$ is defined as

$\delta^{H}(P,R):=\max\left\\{\underset{x\in P}{\sup}\left\\{\underset{y\in
R}{\inf}\|x-y\|\right\\},\underset{y\in R}{\sup}\left\\{\underset{x\in
P}{\inf}\|y-x\|\right\\}\right\\},$

or alternatively by $\delta^{H}(P,R)=\min\left\\{\lambda\geq 0:P\subseteq
R+\lambda B_{2}^{d},R\subseteq P+\lambda B_{2}^{d}\right\\}$. For
$P,R\in\mathcal{B}(d),$ the following is a well-known characterization of
$\delta^{H}$:

$\delta^{H}(P,R)=\underset{x\in\partial
B_{2}^{d}}{\sup}\left|g_{P^{\circ}}\left(x\right)-g_{R^{\circ}}(x)\right|.$

See [32, Theorem 1.8.11]. Throughout the work $\mathcal{K}_{d}$ and
$\mathcal{B}(d)$ will be regarded as metric spaces with respect to the
Hausdorff distance. It is convenient to point out that in
$\otimes_{i=1}^{l}\mathbb{R}^{d_{i}},$ the corresponding Hausdorff distance
will be the one induced by the Euclidean norm $\|\cdot\|_{H}.$ In this case,
the class of $0$-symmetric convex bodies in
$\otimes_{i=1}^{l}\mathbb{R}^{d_{i}},$ denoted by
$\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}),$ will be considered as a
metric space with respect to the Hausdorff distance.

For any $0$-symmetric convex body $P\subset\mathbb{R}^{d}$ or
$P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}},$ the Löwner ellipsoid is denoted
by $Low(P).$ Recall that $Low(P)$ is the ellipsoid of minimal volume
containing $P$ [34, Chapter 3]. We refer the reader to the monograph [32] for
the basics about convex bodies.

###  2.1. The space of tensorial bodies

By $\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ we denote the class of
tensorial bodies in $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}.$ It consists of
$0$-symmetric convex bodies $P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ for
which there exists $0$-symmetric convex bodies $P_{i}$ in the Euclidean space
$\mathbb{R}^{d_{i}}$, $i=1,\ldots,l,$ such that

(2.1) $P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\subseteq P\subseteq
P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}.$

Here, $\otimes_{\pi}$ is the projective tensor product of convex bodies [8,
19]

$P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}:=\text{conv}\left\\{x^{1}\otimes\cdots\otimes
x^{l}\in\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}:x^{i}\in
P_{i},i=1,\ldots,l\right\\},$

and $\otimes_{\epsilon}$ is the injective tensor product of convex bodies
(with zero in the interior) [9, 19]

$P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}:=(P_{1}^{\circ}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}^{\circ})^{\circ}.$

The polarity in $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ is the one determined by
the scalar product $\langle\cdot,\cdot\rangle_{H},$ see (1.3). Alternatively,
the class of tensorial bodies can be described in terms of the Minkowski
functionals. A $0$-symmetric convex body
$P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ is a tensorial body if and only
if

(2.2) $g_{P}\left(x^{1}\otimes\cdots\otimes
x^{l}\right)=g_{P_{1}}\left(x^{1}\right)\cdots
g_{P_{l}}\left(x^{l}\right)\text{ and }$ (2.3)
$g_{P^{\circ}}\left(x^{1}\otimes\cdots\otimes
x^{l}\right)=g_{P_{1}^{\circ}}\left(x^{1}\right)\cdots
g_{P_{l}^{\circ}}\left(x^{l}\right),$

for some $P_{i}\in\mathcal{B}(d_{i})$ and every $x^{i}\in\mathbb{R}^{d_{i}},$
$i=1,\ldots,l.$ Notice that, from the above equalities and the bipolar theorem
follow easily that $P$ is a tensorial body if and only if so is $P^{\circ}$.
Moreover $\lambda P$ is a tensorial body, provided $\lambda\neq 0$ and $P$ is
a tensorial body [18, Proposition 3.5].

The class of tensorial bodies was introduced in [18]. It arises from the
theory of norms on tensor products of Banach spaces. In this sense, in [18,
Proposition 3.1], it was proved that a norm $\alpha(\cdot)$ on the tensor
product $\otimes_{i=1}^{l}(\mathbb{R}^{d_{i}},\|\cdot\|_{i})$ is a reasonable
crossnorm if and only if its closed unit ball
$B_{(\otimes(\mathbb{R}^{d_{i}},\|\cdot\|_{i}),\alpha)}$ is such that

$B_{\|\cdot\|_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{\|\cdot\|_{l}}\subseteq
B_{(\otimes(\mathbb{R}^{d_{i}},\|\cdot\|_{i}),\alpha)}\subseteq
B_{\|\cdot\|_{1}}\otimes_{\epsilon}\cdots\otimes_{\epsilon}B_{\|\cdot\|_{l}},$

where $B_{\|\cdot\|_{i}}\in\mathcal{B}(d_{i})$ is the closed unit ball of the
norm $\|\cdot\|_{i},$ $i=1,\dots,l.$ In particular,
$B_{(\otimes(\mathbb{R}^{d_{i}},\|\cdot\|_{i}),\alpha)}$ is a tensorial body
in $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$. The projective $\otimes_{\pi}$ and
the injective $\otimes_{\epsilon}$ tensor products correspond with the so
called projective $\pi(\cdot)$ and injective $\epsilon(\cdot)$ tensor norms,
by means of the following relations:

(2.4)
$B_{\otimes_{\pi}(\mathbb{R}^{d_{i}},g_{P_{i}})}=P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\thinspace\text{
and
}B_{\otimes_{\epsilon}(\mathbb{R}^{d_{i}},g_{P_{i}})}=P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}.$

Here $B$ denotes the closed unit ball of the projective and the injective
tensor norms, and $P_{i}\subset\mathbb{R}^{d_{i}},$ $i=1,\dots,l,$ is an
arbitrary $l$-tuple of $0$-symmetric convex bodies. In [18, Theorem 3.2], by
means of the Minkowski functional, a bijection between
$\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ and the class of reasonable
crossnorms on finite dimensions is exhibited. In consequence, the tensorial
bodies are, precisely, the closed unit balls of finite-dimensional reasonable
crossnorms. For the notation and a fuller treatment of tensor norms, we refer
the reader to the classical monographs [16, 30]. For a detailed discussion
about the interplay between tensorial bodies and tensor norms, we refer to
[18, 19].

The class of tensorial bodies $\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ is
contained in $\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}),$ and thus it
is also a metric space with respect to the Hausdorff distance. We refer to it
as the space of tensorial bodies. Indeed, it is a $GL_{\otimes}$-space [21,
Section 4]. The action is given by the evaluation $T(P),$ for $T\in
GL_{\otimes}$ and $P\in\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l}).$ In
particular, for the projective $\otimes_{\pi}$ and the injective
$\otimes_{\epsilon}$ tensor product, the following holds:

(2.5)
$T(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l})=T_{1}(P_{\sigma(1)})\otimes_{\alpha}\cdots\otimes_{\alpha}T_{l}(P_{\sigma(l)}),\text{
}\alpha=\pi,\epsilon,$

for every tuple of $0$-symmetric convex bodies
$P_{i}\subset\mathbb{R}^{d_{i}},$ $i=1,\ldots,l.$ In this case, we have used
the representation of $T\in GL_{\otimes}$ as $T=(T_{1}\otimes\cdots\otimes
T_{l})U_{\sigma}$ for some $T_{i}\in GL(d_{i}),$ $i=1,\dots,l,$ and
$U_{\sigma}\in\mathcal{P}$ (see Section 1.2). Many topological properties of
this space were exhibited in [21], below we summarize those which are relevant
for this work.

For every fixed tuple $P_{i}\in\mathcal{B}(d_{i}),$ $i=1,\dots,l,$ the
hyperspace $\mathcal{B}_{P_{1},\dots,P_{l}}(d_{1},\ldots,d_{l})$ consists of
the tensorial bodies for which (2.1) hold. If
$P\in\mathcal{B}_{P_{1},\dots,P_{l}}(d_{1},\ldots,d_{l}),$ we say that $P$ is
a tensorial body with respect to $P_{1},\dots,P_{l}.$ As a consequence of [18,
Corollary 3.4], for every tensorial body $P$ in
$\otimes_{i=1}^{l}\mathbb{R}^{d_{i}},$ one can construct explicitly different
$l$-tuples
$(P_{1},\ldots,P_{l})\in\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})$
for which $P\in\mathcal{B}_{P_{1},\dots,P_{l}}(d_{1},\ldots,d_{l}).$ In this
sense, to simplify the arguments in the forthcoming proofs, it is convenient
to fix one of these $l$-tuples associated to each
$P\in\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l}).$

######  Remark 2.1.

Let $P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ be a tensorial body and let
$P^{i}\in\mathcal{B}(d_{i}),$ $i=1,\ldots,l,$ be defined as

$P^{i}:=\left\\{x^{i}\in\mathbb{R}^{d_{i}}:\frac{e^{d_{1}}_{1}\otimes\cdots\otimes
e^{d_{i-1}}_{1}\otimes x^{i}\otimes e^{d_{i+1}}_{1}\otimes\cdots\otimes
e^{d_{l}}_{1}}{g_{P}(e^{d_{1}}_{1}\otimes e^{d_{2}}_{1}\otimes\cdots\otimes
e^{d_{l}}_{1})}\in P\right\\},$

for $i=1,\dots,l-1,$ and for $i=l$ as
$P^{l}:=\left\\{x^{l}\in\mathbb{R}^{d_{l}}:e^{d_{1}}_{1}\otimes\cdots\otimes
e^{d_{l-1}}_{1}\otimes x^{l}\in P\right\\}.$ It is straightforward to check
that $P^{i}$ is precisely the $0$-symmetric convex body
$P_{i}^{e_{1}^{d_{1}},e_{1}^{d_{2}},\ldots,\lambda e_{1}^{d_{l}}},$
$i=1,\ldots,l,$ $\lambda=\frac{1}{g_{P}(e^{d_{1}}_{1}\otimes
e^{d_{2}}_{1}\otimes\cdots\otimes e^{d_{l}}_{1})},$ defined in [18, (9)].
Thus, from [18, Corollary 3.4], it follows directly that $P$ is a tensorial
body with respect to $P^{1},\dots,P^{l}.$

By $\mathcal{E}_{\otimes}(d_{1},\ldots,d_{l}),$ we denote the class of
$0$-symmetric ellipsoids in $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ which also
are tensorial bodies. Recall that
$E\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ is a $0$-symmetric ellipsoid
(not necessarily tensorial) if there exists a linear isomorphism $T$ on
$\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ such that
$E=T(B_{2}^{d_{1},\dots,d_{l}}).$ We usually write $\mathcal{E}_{\otimes}$
instead of $\mathcal{E}_{\otimes}(d_{1},\ldots,d_{l}).$

######  Proposition 2.2.

[21] The following hold:

  1. (1)

The tensor products $\otimes_{\pi},$ $\otimes_{\epsilon}$:
$\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})\rightarrow\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$
are continuous with respect to the Hausdorff distance.

  2. (2)

The map
$conv_{\otimes}:\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})\rightarrow\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l}),$
defined as

(2.6) $conv_{\otimes}(P):=conv(P\cap\Sigma_{d_{1},\ldots,d_{l}}),$

is a $GL_{\otimes}$-equivariant retraction onto
$\Pi(d_{1},\dots,d_{l}):=\\{P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}:P_{i}\in\mathcal{B}(d_{i})\\}.$

  3. (3)

The map $\ell_{\otimes}=Low\circ conv_{\otimes}$ sending $P$ to
$Low(conv_{\otimes}(P))$ is a $GL_{\otimes}$-equivariant retraction of
$\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ onto
$\mathcal{E}_{\otimes}(d_{1},\ldots,d_{l}).$

######  Theorem 2.3.

[21] Let $p=\frac{d_{1}(d_{1}+1)}{2}+\cdots+\frac{d_{l}(d_{l}+1)}{2},$ with
$d_{i}\geq 2$ for $i=1,\dots,l.$ The following hold:

  1. (1)

$\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ is a closed and contractible
subspace of $\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}).$

  2. (2)

$\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ is a proper $GL_{\otimes}$-space.

  3. (3)

$\mathcal{E}_{\otimes}(d_{1},\ldots,d_{l})$ is the $GL_{\otimes}$-orbit of
$B_{2}^{d_{1},\dots,d_{l}}.$ Furthermore, it is homeomorphic to
$\mathbb{R}^{p}.$

  4. (4)

The set $\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l}),$ defined as

$\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l}):=\\{P\in\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l}):\ell_{\otimes}(P)=B_{2}^{d_{1},\ldots,d_{l}}\\},$

is a compact $O_{\otimes}$-global slice for the $GL_{\otimes}$-space
$\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l}).$ Moreover, the spaces
$\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})/GL_{\otimes}(d_{1},\ldots,d_{l})$
and
$\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})/O_{\otimes}(d_{1},\ldots,d_{l})$
are homeomorphic.

  5. (5)

The orbit space
$\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})/GL_{\otimes}(d_{1},\ldots,d_{l})$
is compact and metrizable. Indeed, the logarithm $\text{
log}\delta_{\otimes}^{BM}$ of the tensorial Banach-Mazur distance
$\delta_{\otimes}^{BM}$ is a metric on it. The tensorial Banach-Mazur distance
$\delta_{\otimes}^{BM}$ is defined as

(2.7) $\delta_{\otimes}^{BM}\left(P,R\right):=\inf\left\\{\lambda\geq
1:R\subseteq TP\subseteq\lambda R,\text{ for }T\in
GL_{\otimes}(d_{1},\dots,d_{l})\right\\},$

for $P,R\in\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l}).$

By $\mathcal{E}(d_{i}),$ we denote the class of $0$-symmetric ellipsoids in
$\mathbb{R}^{d_{i}},$ $i=1,\ldots,l.$ Given any $l$-tuple of $0$-symmetric
ellipsoids $E_{i}=T_{i}(B_{2}^{d_{i}}),$ $T_{i}\in GL(d_{i})$ and
$i=1,\dots,l,$ their Hilbertian tensor product
$E_{1}\otimes_{2}\cdots\otimes_{2}E_{l}$ is a tensorial body in
$\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ defined as

$E_{1}\otimes_{2}\cdots\otimes_{2}E_{l}:=T_{1}\otimes\cdots\otimes
T_{l}(B_{2}^{d_{1},\ldots,d_{l}}).$

It is important to notice that this definition does not depend on the
selection of the maps $T_{i}.$ See [8] or [18, Section 4] for the details and
the basic properties of $\otimes_{2}$. In [18, Corollary 4.3], it is proved
that the class of tensorial ellipsoids
$\mathcal{E}_{\otimes}(d_{1},\ldots,d_{l})$ consists only of tensor products
$E_{1}\otimes_{2}\cdots\otimes_{2}E_{l}$ of ellipsoids
$E_{i}\in\mathcal{E}(d_{i}),$ $i=1,\ldots,l$. Actually, the Euclidean ball
$B_{2}^{d_{1},\dots,d_{l}}$ is a tensorial ellipsoid, since
$B_{2}^{d_{1},\dots,d_{l}}=I_{1}\otimes\cdots\otimes
I_{l}(B_{2}^{d_{1},\dots,d_{l}}),$ where $I_{i}$ is the identity map on
$\mathbb{R}^{d_{i}},$ $i=1,\dots,l.$ In particular, it is a tensorial body
with respect to the Euclidean balls $B_{2}^{d_{1}},\dots,B_{2}^{d_{l}}.$

######  Proposition 2.4.

The tensor product
$\otimes_{2}:\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l})\rightarrow\mathcal{E}_{\otimes}(d_{1},\ldots,d_{l})$
is continuous with respect to the Hausdorff distance.

###### Proof.

Recall that the topology on $GL(d_{i})$ is the one inherited from the space of
linear maps on the Euclidean space $\mathbb{R}^{d_{i}},$ similarly the
topology on $GL_{\otimes}$ is the one determined by the space of linear maps
on $\otimes_{H,i=1}^{l}\mathbb{R}^{d_{i}}.$ We will need the topological
representative of $\mathcal{E}(d_{i})$ exhibited in the proof of [5, Corollary
3.9]. There, $\mathcal{E}(d_{i})$ is represented as the closed subset
$\mathcal{A}_{i}\subset GL(d_{i})$ consisiting of the positive self-adjoint
maps $T_{i}:\mathbb{R}^{d_{i}}\rightarrow\mathbb{R}^{d_{i}}.$ The
homeomorphism is given by the map
$\xi_{i}:\mathcal{E}(d_{i})\rightarrow\mathcal{A}_{i}$ sending
$E_{i}\in\mathcal{E}(d_{i})$ to the unique linear map
$\xi_{i}(E_{i})\in\mathcal{A}_{i}$ such that
$\xi_{i}(E_{i})B_{2}^{d_{i}}=E_{i}$, see [5, (3.4)]. If we let
$F:\mathcal{A}_{1}\times\cdots\times\mathcal{A}_{l}\rightarrow\mathcal{E}_{\otimes}(d_{1},\ldots,d_{l})$
be the map

$F(T_{1},\ldots,T_{l})=T_{1}\otimes\cdots\otimes
T_{l}(B_{2}^{d_{1},\ldots,d_{l}}),$

then $\otimes_{2}=F\circ(\xi_{1},\ldots,\xi_{l}).$ Thus, for any
$S_{i},T_{i}\in\mathcal{A}_{i}$

$\delta^{H}(F(T_{1},\ldots,T_{l}),F(S_{1},\ldots,S_{l}))\leq\|T_{1}\otimes\cdots\otimes
T_{l}-S_{1}\otimes\cdots\otimes S_{l}\|,$

where $\|T_{1}\otimes\cdots\otimes T_{l}-S_{1}\otimes\cdots\otimes S_{l}\|$
denotes the usual norm of linear maps on the Euclidean space
$\otimes_{H,i=1}^{l}\mathbb{R}^{d_{i}}.$ This shows that $F$ is a continuous
map and so the same holds for $\otimes_{2}$. ∎

The following lemmas will be used later in the work. Our proof of Lemma 2.6
follows along the same ideas of [25, Theorem 2.2], we have included it for the
sake of transparency.

######  Lemma 2.5.

The map $\nu:\mathcal{B}(d)\times\mathcal{B}(d)\rightarrow(0,\infty),$ $d\geq
2,$ defined as $\nu(P,Q)=\sup_{z\in P}g_{Q}(z)$ is continuous.

###### Proof.

It can be easily checked that if $P,R,\bar{P},\bar{R}\in\mathcal{B}(d),$ then

$\displaystyle|\nu(P,R)-\nu(\bar{P},\bar{R})|$
$\displaystyle\leq|\nu(P,R)-\nu(P,\bar{R})|+|\nu(P,\bar{R})-\nu(\bar{P},\bar{R})|$
$\displaystyle\leq\sup_{z\in
P}|g_{R}(z)-g_{\bar{R}}(z)|+\delta^{H}(P,\bar{P})\nu(B_{2}^{d},\bar{R})$

Now, for every pair of sequences
$\\{P_{n}\\},\\{R_{n}\\}\subset\mathcal{B}(d)$ converging to $P,R$
respectively, we have that $\\{\nu(B_{2}^{d},R_{n})\\}_{n}$ is bounded and
that $\\{g_{R_{n}}(\cdot)\\}_{n}$ converges uniformly on $P$ to $g_{R}(\cdot)$
([21, Lemma 3.1]). Thus, from the above inequalities, it follows that
$\nu(P_{n},R_{n})$ approaches to $\nu(P,R)$ and so the map $\nu$ is
continuous. ∎

######  Lemma 2.6.

Let $A$ and $K$ be $0$-symmetric convex bodies in $\mathbb{R}^{d}$ with
$A\subsetneqq K.$ Then the hyperspace
$csb_{A}(K):=\\{P\in\mathcal{B}(d):A\subseteq P\subseteq K\\}$ is homeomorphic
to $\mathcal{Q}$.

###### Proof.

First, let us observe that $csb_{A}(K)$ is compact. Indeed, by the Blaschke
selection theorem, any sequence $\\{R_{n}\\}_{n}\subset csb_{A}(K)$ has a
subsequence $\\{R_{n_{i}}\\}$ converging to a compact convex set $R.$ Clearly,
$R$ must be $0$-symmetric and $A\subseteq R\subseteq K.$ Therefore $R\in
csb_{A}(K)$ and the hyperspace is compact.

On the other hand, the support function provides an affine embedding of
$csb_{A}(K)$ into $C(\mathbb{S}^{d-1}),$ the space of continuous functions on
the sphere $\mathbb{S}^{d-1}=\partial B_{2}^{d}.$ By the properties of the
Minkowski sum, the image of $csb_{A}(K)$ under this embedding is convex.

Hence, $csb_{A}(K)$ is homeomorphic to a compact convex subset of a Banach
space. Therefore, by [13, Theorem 7.1], $csb_{A}(K)$ is homeomorphic to either
$\mathcal{Q}$ or to $[0,1]^{n}$ for some natural number $n$. Since every cube
$[0,1]^{n}$ has covering dimension equal to $n$, in order to complete the
proof it is enough to show that $csb_{A}(K)$ does not have finite covering
dimension. To that end, we will show that $csb_{A}(K)$ contains an $n$-cell
for every $n\geq 1,$ the result then will follow from [29, Ch. 3, Proposition
1.5]).

Since $A\neq K,$ there exists $x_{0}\in\text{int}(K)\setminus A.$ Let
$r_{0}>0$ be such that the closed unit ball
$B[x_{0},r_{0}]:=x_{0}+r_{0}B_{2}^{d}\subset\text{int}(K)\setminus A$ and
$2r_{0}B_{2}^{d}\subset A$. Let us denote by $p(x_{0},A)\in A$ the point such
that $\text{dist}(x_{0},A)=\|x_{0}-p(x_{0},A)\|,$ and let

$u(x_{0},A)=\frac{x_{0}-p(x_{0},A)}{\text{dist}(x_{0},A)}.$

It is known that $H,$ the hyperplane through $p(x_{0},A)$ orthogonal to
$u(x_{0},A),$ supports $A$ ([32, Lemma 1.3.1]. Consider any fixed $v\in H$
with $v\neq p(x_{0},A),$ and let $\mathcal{V}$ be the 2-dimensional plane
through $p(x_{0},A)$ generated by $v-p(x_{0},A)$ and $u(x_{0},A).$ Namely,

$\mathcal{V}=\\{p(x_{0},A)+s(v-p(x_{0},A))+tu(x_{0},A):s,t\in\mathbb{R}\\}.$

Notice that for every $0<r<r_{0},$ the 2-dimensional closed disk
$D=B[x_{0},r]\cap\mathcal{V}\subset K$ is separated from the section
$A\cap\mathcal{V}$ by $H\cap\mathcal{V}.$ Let $y_{N}=x_{0}+ru(x_{0},A)\in D.$

Claim 1. There exist $\varepsilon>0$ such that for any pair of points $z,y\in
B[y_{N},\varepsilon]\cap\partial D$, the line determined by $z$ and $y$ does
not intersect $A$.

Suppose that the opposite occurs, then for every $\varepsilon>0,$ there exists
$y_{\varepsilon},z_{\varepsilon}\in B[y_{N},\varepsilon]\cap\partial D$ for
which $L_{\varepsilon}\cap A\neq\emptyset,$ where $L_{\varepsilon}$ denotes
the line determined by $z_{\varepsilon}$ and $y_{\varepsilon}$. Pick a point
$a(\varepsilon)\in L_{\varepsilon}\cap A$, and let $L$ be the line in
$\mathcal{V}$ which is tangent to $D$ at the point $y_{N}$. Also, denote by
$\Lambda$ the line generated by $p(x_{0},A)$ and $v$ (namely,
$\Lambda=H\cap\mathcal{V})$. Observe that $L$ and $\Lambda$ are parallel lines
and both are completely contained in the plane $\mathcal{V}$. Since
$L_{\varepsilon}$ intersects $A\cap\mathcal{V}$, and $A\cap\mathcal{V}$ is
contained in the half plane determined by $\Lambda$ that does not contain the
pair $\\{y_{\varepsilon},z_{\varepsilon}\\}$, we also have that
$L_{\varepsilon}\cap\Lambda\neq\emptyset$. Let us denote by $b(\varepsilon)$
the intersection point of these two lines. Consider the triangle in
$\mathcal{V}$ with vertices $y_{\varepsilon},$ $p(x_{0},A)$ and
$a(\varepsilon)$. Observe that $b(\varepsilon)$ lies in the segment with
extremes $y_{\varepsilon}$ and $a(\varepsilon)$ and therefore

$\displaystyle diam(A)$
$\displaystyle\geq\|a(\varepsilon)-p(x_{0},A)\|\geq\|a(\varepsilon)-y_{\varepsilon}\|-\|y_{\varepsilon}-p(x_{0},A)\|$
$\displaystyle\geq\|a(\varepsilon)-y_{\varepsilon}\|-(\|y_{\varepsilon}-x_{0}\|+\|x_{0}-p(x_{0},A)\|)$
$\displaystyle\geq\|b(\varepsilon)-y_{\varepsilon}\|-(r+\text{dist}(x_{0},A))$

Notice that if $\varepsilon$ approaches to $0$, then
$\|b(\varepsilon)-y_{\varepsilon}\|$ diverges to $\infty$. This contradicts
the fact that $A$ is compact and therefore the claim is proved.

Let $\varepsilon\in(0,r)$ be as in Claim 1, and let us denote by $T$ the
planar region determined by the arc $\partial D\cap B[y_{N},\varepsilon]$ and
the chord through its extreme points. Then, for any integer $n\geq 3,$ there
exists an integer $m=m(n)>n$ such that for some regular $m$-sided polygon
$P_{m}$ with sides $S_{1},\ldots,S_{m}$ inscribed in $D,$ there are (at least)
$n$ consecutive sides contained in $T.$ Let us suppose, without loss of
generality, that the sides $S_{1},\ldots,S_{n}$ are contained in $T.$ Next,
for every $j=1,\ldots,n,$ let $C_{j}$ be the segment
$[b_{j},c_{j}]:=\\{tc_{j}+(1-t)b_{j}:t\in[0,1]\\},$ where $b_{j}$ is the
middle point of $S_{j},$ and $c_{j}$ is the intersection of $\partial D$ with
the ray $R_{j}=\\{sb_{j}+(1-s)x_{0}:s\geq 1\\}$ (see Figure 1). By our
selection of $T,$ if $j=1,\ldots,n-1,$ $z\in C_{j}$ and $w\in C_{j+1},$ then
the line $\mathcal{W}=\\{z+t(z-w):t\in\mathbb{R}\\}$ is completely contained
in $\mathcal{V}\setminus A.$

Figure 1. Construction of Lemma 2.6. Here, the sides $S_{1},S_{2},S_{3}$ of
the polygon $P_{8}$ and segments $C_{1},C_{2},C_{3}$ are contained in the
shaded region $T.$

Claim 2. The map $G:C_{1}\times\cdots\times C_{n}\rightarrow csb_{A}(K),$
sending the tuple $(z_{1},\ldots,z_{n})$ to $\text{conv}(\\{\pm
z_{1},\ldots,\pm z_{n}\\}\cup A)$ is a homeomorphism of the $n$-cell
$C_{1}\times\cdots\times C_{n}$ into $csb_{A}(K).$

Clearly, $G(z_{1},\ldots,z_{n})\subseteq K$ is a $0$-symmetric convex body
containing $A.$ In addition, it is not difficult to check that

$\delta^{H}(G(z_{1},\ldots,z_{n}),G(w_{1},\ldots,w_{n}))\leq\sum_{j=1}^{n}\|z_{i}-w_{i}\|.$

Hence, $G$ is a continuous map. Moreover, since $G$ is defined on a compact
set, it must be closed. To prove that $G$ is one-to-one, notice that if
$G(z_{1},\ldots,z_{n})=G(w_{1},\ldots,w_{n}),$ then
$G(z_{1},\ldots,z_{n})\cap\mathcal{V}=G(w_{1},\ldots,w_{n})\cap\mathcal{V}$
and therefore their extreme points coincide. The fact that $2rB_{2}^{d}\subset
2r_{0}B_{2}^{d}\subset A$, in combination with our construction, guarantees
that $\\{z_{1},\dots,z_{m}\\}$ and $\\{w_{1},\dots,w_{m}\\}$ are extreme
points of $G(z_{1},\ldots,z_{n})\cap\mathcal{V}$ and
$G(w_{1},\ldots,w_{n})\cap\mathcal{V}$, respectively. Thus, $z_{j}=w_{j}$ for
every $j=\\{1,\dots,n\\}$ and therefore $G$ is a homeomorphism, as desired.

Notice that Claim 2 implies that $csb_{A}(K)$ contains a $n$-cell, for every
$n\geq 3$. This proves that $csb_{A}(K)$ does not have finite covering
dimension and therefore the proof is complete.

∎

##  3\. A hyperspace associated to $\otimes_{\pi}$

In this section, we calculate the topological structure of the space
$\Pi(d_{1},\dots,d_{l})$ consisting of the tensor products
$P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}$ of $0$-symmetric convex bodies
$P_{i}\subset\mathbb{R}^{d_{i}},$ $i=1,\dots,l.$ To this end, we go further
into the properties of the $GL_{\otimes}$-map
$conv_{\otimes}:\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})\rightarrow\Pi(d_{1},\dots,d_{l}),$
defined in Proposition 2.2.

Let us denote by $H$ the closed subgroup of $GL(d_{1})\times\cdots\times
GL(d_{l})$ defined as

$H:=\\{(\lambda_{1}I_{d_{1}},\ldots,\lambda_{l}I_{d_{l}}):\lambda_{1}\cdots\lambda_{l}=1\text{
and each }\lambda_{i}>0\\}.$

Here $I_{d_{i}}$ denotes the identity map of $\mathbb{R}^{d_{i}},$
$i=1,\dots,l.$ If we consider the natural action of each $GL(d_{i})$ on
$\mathcal{B}(d_{i}),$ then the evaluation
$(T_{1}(P_{1}),\ldots,T_{l}(P_{l})),$ $T_{i}\in GL(d_{i})$ and
$P_{i}\in\mathcal{B}(d_{i}),$ $i=1,\dots,l,$ defines a continuous action of
$GL(d_{1})\times\cdots\times GL(d_{l})$ on
$\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l}).$ As a consequence,
its restriction to $H$ determines a continuous action of $H$ on
$\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l}).$ In a similar way, in
the subspace $\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l}),$
consisting of $l$-tuples of $0$-symmetric ellipsoids, the restriction of the
action on $\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})$ also
induces a natural action of $H$ on
$\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l}).$

We now exhibit the relation between the $H$-orbits of the $l$-tuples
$(P_{1},\dots,P_{l})\in\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})$
and $\otimes_{\pi}.$ By [19, Proposition 5], for every $l$-tuple
$P_{i}\in\mathcal{B}(d_{i}),$ $i=1,\dots,l,$ we have that

(3.1)
$\displaystyle\lambda_{1}P_{1}\otimes_{\pi}\cdots\otimes_{\pi}\lambda_{l}P_{l}=(\lambda_{1}\cdots\lambda_{l})P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}=P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l},$

as long as each $\lambda_{i}>0$ and $\lambda_{1}\cdots\lambda_{l}=1.$ Hence,
$\lambda_{1}P_{1}\otimes_{\pi}\cdots\otimes_{\pi}\lambda_{l}P_{l}$ belongs to
$\otimes_{\pi}^{-1}(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}).$ Conversely,
if
$(R_{1},\ldots,R_{l})\in\otimes_{\pi}^{-1}(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}),$
then there exists real numbers $\lambda_{i}>0$ such that
$\lambda_{1}\cdots\lambda_{l}=1$ and $R_{i}=\lambda_{i}P_{i},$ $i=1,\ldots,l$
[18, Proposition 3.6]. These inclusions prove that

(3.2)
$\otimes_{\pi}^{-1}(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l})=\\{(\lambda_{1}P_{1},\dots,\lambda_{l}P_{l}):\lambda_{1}\cdots\lambda_{l}=1\text{
and each }\lambda_{i}>0\\}.$

In consequence,
$\otimes_{\pi}^{-1}(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l})$ is the
$H$-orbit of each $l$-tuple
$(P_{1},\dots,P_{l})\in\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l}).$
To shorten notation, we usually write $\mathcal{B}_{\otimes}$ and $\Pi$
instead of $\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ and
$\Pi(d_{1},\ldots,d_{l}).$

######  Proposition 3.1.

The following hold:

  1. (1)

The $GL_{\otimes}$-equivariant retraction $conv_{\otimes}$ is proper.

  2. (2)

$\Pi(d_{1},\ldots,d_{l})$ is a $GL_{\otimes}$-strong deformation retract of
$\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l}).$

###### Proof.

1\. We show that for any compact subset $Y\subseteq\Pi,$
$conv_{\otimes}^{-1}(Y)$ is compact too. To this end, let $\\{P_{n}\\}_{n}$ be
a sequence in $conv_{\otimes}^{-1}(Y)$, we will see that it contains a
convergent subsequence. By the compactness of $Y,$ there exists a subsequence
$\\{P_{n_{k}}\\}$ such that $conv_{\otimes}(P_{n_{k}})$ converges to some
$P\in Y.$ Now, from the triangle inequality we have

(3.3)
$\delta^{H}(P_{n_{k}},P)\leq\delta^{H}(P_{n_{k}},conv_{\otimes}(P_{n_{k}}))+\delta^{H}(conv_{\otimes}(P_{n_{k}}),P).$

On the other hand, if $P_{n_{k}}^{i},$ $i=1,\dots,l,$ are the convex bodies
associated to $P_{n_{k}}$ in Remark 2.1, then $P_{n_{k}}$ and
$conv_{\otimes}(P_{n_{k}})$ belong to
$\mathcal{B}_{P_{n_{k}}^{1},\dots,P_{n_{k}}^{l}}(d_{1},\dots,d_{l}),$ and,
from (2.1), we have

(3.4)
$\displaystyle\delta^{H}(P_{n_{k}},conv_{\otimes}(P_{n_{k}}))\leq\delta^{H}(P_{n_{k}}^{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{n_{k}}^{l},P_{n_{k}}^{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{n_{k}}^{l}).$

Additionally, from the relations between the projective and the injective norm
exhibited in [17, Proposition 2.4] combined with the identities in (2.4), we
get

$P_{n_{k}}^{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{n_{k}}^{l}\subset
P_{n_{k}}^{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{n_{k}}^{l}\subset\left(\prod_{i=1}^{l-1}d_{i}\right)P_{n_{k}}^{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{n_{k}}^{l},$

and so

$\displaystyle\delta^{H}(P_{n_{k}}^{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{n_{k}}^{l},$
$\displaystyle
P_{n_{k}}^{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{n_{k}}^{l})\leq\left(\prod_{i=1}^{l-1}d_{i}-1\right)\nu(P_{n_{k}}^{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{n_{k}}^{l},B_{2}^{d_{1},\dots,d_{l}})$
(3.5)
$\displaystyle\overset{*}{=}\left(\prod_{i=1}^{l-1}d_{i}-1\right)\nu(conv_{\otimes}(P_{n_{k}}),B_{2}^{d_{1},\dots,d_{l}}).$

(*) Folllows from the fact that $conv_{\otimes}$ is constant on
$\mathcal{B}_{P_{n_{k}}^{1},\dots,P_{n_{k}}^{l}}(d_{1},\dots,d_{l})$ [21,
Proposition 4.4], and thus
$conv_{\otimes}(P_{n_{k}})=P_{n_{k}}^{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{n_{k}}^{l}.$

Therefore, from the convergence of $conv_{\otimes}(P_{n_{k}})$ to $P$, the
continuity of $\nu$ (Lemma 2.5), and the inequalities (3.3), (3.4) and (3), it
follows that $\\{P_{n_{k}}\\}_{n_{k}}$ must be bounded. Hence, by the Blaschke
selection theorem [32, Theorem 1.8.6], we may suppose (without lost of
generality) that $\\{P_{n_{k}}\\}_{n_{k}}$ converges to some $0$-symmetric
compact convex set $R\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}.$ Indeed,
since $conv_{\otimes}(P_{n_{k}})\subseteq P_{n_{k}},$ then $P\subset R$ and so
$R$ is a $0$-symmetric convex body. Finally, since
$\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ is closed in
$\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}})$ [21, Proposition 3.4], we
know that $R\in\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l}).$ This proves that
$conv_{\otimes}^{-1}(Y)$ is compact, as desired.

2\. Let
$W:\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})\times[0,1]\rightarrow\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$
be defined as

$W(P,t)=(1-t)P+tconv_{\otimes}(P)$

We will show that $W$ is the desired $GL_{\otimes}$-homotopy. Let $P^{i},$
$i=1,\dots,l,$ be the convex bodies associated to $P$ in Remark 2.1. Notice
that $W$ is well-defined since $P$ and $conv_{\otimes}(P)$ belong to
$\mathcal{B}_{P^{1},\dots,P^{l}}(d_{1},\dots,d_{l})$ which is convex [21,
Proposition 3.4]. The continuity and equivariance of $W$ follow directly from
those of $conv_{\otimes}.$ Also, $W(P,0)=P$ and $W(P,1)=conv_{\otimes}(P),$
for every tensorial body $P.$ Furthermore, if $P\in\Pi,$ then
$conv_{\otimes}(P)=P$ and

$W(P,t)=(1-t)P+tconv_{\otimes}(P)=(1-t)P+tP=P,$

for any $t\in[0,1].$ This finishes the proof.

∎

######  Theorem 3.2.

Let $d_{i}\geq 2,$ $i=1,\dots,l,$ and
$p=\frac{d_{1}(d_{1}+1)+\cdots+d_{l}(d_{l}+1)}{2}.$ The following hold:

  1. (1)

$\Pi(d_{1},\ldots,d_{l})$ is homeomorphic to the orbit space
$\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})/H.$

  2. (2)

The space of tensorial ellipsoids $\mathcal{E}_{\otimes}(d_{1},\ldots,d_{l})$
is homeomorphic to the orbit space
$\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l})/H.$ In paticular, the
latter is homeomorphic to $\mathbb{R}^{p}.$

  3. (3)

$\Pi(d_{1},\ldots,d_{l})$ is homeomorphic to
$\mathcal{Q}\times\mathbb{R}^{p}.$

###### Proof.

1\. Recall that $\otimes_{\pi}$ is continuous, see [21, Proposition 3.3].
This, together with (3.2), shows that $\otimes_{\pi}$ induces a continuous
bijective map
$\tilde{\otimes}_{\pi}:\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})/H\rightarrow\Pi$
such that $\otimes_{\pi}=\tilde{\otimes}_{\pi}\circ\rho,$ where
$\rho:\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})\rightarrow\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})/H$
is the orbital projection. We will show that $\tilde{\otimes}_{\pi}$ is indeed
a homeomorphism. To this end, let
$W\subseteq\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})$ be a closed
subset such that $W=\bigcup\\{\rho(P_{1},\dots,P_{l}):(P_{1},\dots,P_{l})\in
W\\},$ we will see that $\otimes_{\pi}(W)$ is closed too. Suppose that
$\\{P_{n}\\}_{n}\subset\otimes_{\pi}(W)$ is a sequence converging to some
$P\in\Pi,$ and let $P_{n}^{i}\subset\mathbb{R}^{d_{i}},$ $i=1,\dots,l,$ be the
$0$-symmetric convex bodies associated to $P_{n}$ in Remark 2.1. Since
$P_{n}=P_{1n}\otimes_{\pi}\cdots\otimes_{\pi}P_{ln}$ for some
$(P_{1n},\ldots,P_{ln})\in W,$ then
$P_{n}\in\mathcal{B}_{P_{1n},\ldots,P_{ln}}\cap\mathcal{B}_{P_{n}^{1},\ldots,P_{n}^{l}}$
and, by [18, Proposition 3.6], $P_{n}^{i}=\lambda_{n}^{i}P_{in}$ for some
$\lambda_{n}^{i}>0$ such that $\lambda_{n}^{1}\cdots\lambda_{n}^{l}=1.$ So,
$(P_{n}^{1},\ldots,P_{n}^{l})\in W.$ Using the fact that $P_{n}^{i}$ converges
to $P^{i}$ (see [21, Lemma 3.1 and Proposition 3.7]), we know that
$P_{n}^{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{n}^{l}$ approaches to
$P^{1}\otimes_{\pi}\cdots\otimes_{\pi}P^{l}$ and the latter belongs to
$\otimes_{\pi}(W).$ Furthermore, since

$\displaystyle
P_{n}^{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{n}^{l}=\lambda_{n}^{1}P_{1n}\otimes_{\pi}\cdots\otimes_{\pi}\lambda_{n}^{l}P_{ln}=P_{1n}\otimes_{\pi}\cdots\otimes_{\pi}P_{ln}=P_{n},$

then $P=P^{1}\otimes_{\pi}\cdots\otimes_{\pi}P^{l}\in\otimes_{\pi}(W).$ This
shows that $\otimes_{\pi}(W)$ is closed, as desired.

2\. This proof follows along the same ideas of 1. However, we have included it
for the sake of completeness. Notice that the restriction $\rho_{|}$ of the
orbit map $\rho$ to $\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l})$
is precisely the orbital projection onto
$\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l})/H.$ In addition, by
[19, Proposition 5], for any $l$-tuple $E_{i}\in\mathcal{E}(d_{i}),$
$i=1,\ldots,l,$ we have that

$\displaystyle\lambda_{1}E_{1}\otimes_{2}\cdots\otimes_{2}\lambda_{l}E_{l}=(\lambda_{1}\cdots\lambda_{l})E_{1}\otimes_{2}\cdots\otimes_{2}E_{l}=E_{1}\otimes_{2}\cdots\otimes_{2}E_{l},$

whenever each $\lambda_{i}>0$ and $\lambda_{1}\cdots\lambda_{l}=1.$ In
consequence, the $H$-orbit $H((E_{1},\ldots,E_{l}))$ is such that
$\otimes_{2}[H((E_{1},\ldots,E_{l}))]=\\{E_{1}\otimes_{2}\cdots\otimes_{2}E_{l}\\}.$
This, along with Proposition 2.4, yield that $\otimes_{2}$ determines a
continuous injective map
$\tilde{\otimes}_{2}:\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l})/H\rightarrow\mathcal{E}_{\otimes}(d_{1},\ldots,d_{l})$
such that $\otimes_{2}=\tilde{\otimes}_{2}\circ\rho_{|}.$ Moreover, given that
in [18, Corollary 4.3], it is proved that $\mathcal{E}_{\otimes}$ consists
only of Hilbertian tensor products of ellipsoids $E_{i}\in\mathcal{E}(d_{i}),$
$i=1,\ldots,l$, then $\tilde{\otimes}_{2}$ is a surjective map.

We now turn to prove that $\tilde{\otimes}_{2}$ is a homeomorphim. To this
end, we will consider a closed subset
$W\subseteq\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l})$ such that
$W=\bigcup\\{\rho_{|}(E_{1},\ldots,E_{l}):(E_{1},\ldots,E_{l})\in W\\},$ and
we shall prove that $\otimes_{2}(W)$ is a closed set. Let
$\\{E_{n}\\}_{n}\subseteq\otimes_{2}(W)$ be a sequence converging to
$E\in\mathcal{E}_{\otimes}.$ Clearly
$E_{n}=E_{1n}\otimes_{2}\cdots\otimes_{2}E_{ln},$ for some
$(E_{1n},\ldots,E_{ln})\in W.$ Also, by [18, Proposition 3.6] and the fact
that $E_{n}$ belongs to
$\mathcal{B}_{E_{n}^{1},\ldots,E_{n}^{l}}\cap\mathcal{B}_{E_{1n},\ldots,E_{ln}},$
it follows that $E_{n}^{i}=\lambda_{n}^{i}E_{in},$ with $\lambda_{n}^{i}>0$
and $\lambda_{n}^{1}\cdots\lambda_{n}^{l}=1.$ Thus,
$(E_{n}^{1},\ldots,E_{n}^{l})\in W$ and, since $E_{n}^{i}$ converges to
$E^{i}$ (see [21, Lemma 3.1 and Proposition 3.7]), we have that
$(E^{1},\ldots,E^{l})\in W.$ Indeed, by the continuity of $\otimes_{2}$
(Proposition 2.4), we also know that
$E_{n}^{1}\otimes_{2}\cdots\otimes_{2}E_{n}^{l}$ approaches to
$E^{1}\otimes_{2}\cdots\otimes_{2}E^{l}.$ Finally, since

$\displaystyle
E_{n}^{1}\otimes_{2}\cdots\otimes_{2}E_{n}^{l}=\lambda_{n}^{1}E_{1n}\otimes_{2}\cdots\otimes_{2}\lambda_{n}^{l}E_{ln}=E_{1n}\otimes_{2}\cdots\otimes_{2}E_{ln}=E_{n},$

then $E_{n}^{1}\otimes_{2}\cdots\otimes_{2}E_{n}^{l}$ converges to $E$ and
$E=E^{1}\otimes_{2}\cdots\otimes_{2}E^{l}\in\otimes_{2}(W).$ This proves that
$\otimes_{2}(W)$ is closed as required. The last part of the result now
follows from the fact that $\mathcal{E}_{\otimes}(d_{1},\ldots,d_{l})$ is
homeomorphic to $\mathbb{R}^{p}$ [21, Corollary 4.2].

3\. Let $L(d_{i})\subset\mathcal{B}(d_{i})$ be the subspace consisting of
$P\in\mathcal{B}(d_{i})$ such that $Low(P)=B_{2}^{d_{i}}.$ We shall prove that
$\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})$ is homeomorphic to
the product $L(d_{1})\times\cdots\times
L(d_{l})\times\left(\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l})/H\right).$
The result then will be a consequence of 2., and the fact that $L(d_{i})$ is
homeomorphic to $\mathcal{Q}$ [2, Theorem 1.4], and
$\mathcal{Q}\times\mathcal{Q}$ is homeomorphic to $\mathcal{Q}$ (see [24,
Excercise 2, §1.1]). We will require some notation from [5]. Let
$\ell_{i}:\mathcal{B}(d_{i})\rightarrow\mathcal{E}(d_{i})$ be the
$GL(d_{i})$-map sending each $P_{i}\in\mathcal{B}(d_{i})$ to its Löwner
ellipsoid $\ell_{i}(P_{i})=Low(P_{i}),$ and let
$r_{i}:\mathcal{B}(d_{i})\rightarrow L(d_{i})$ be the $O(d_{i})$-retraction of
[5, Corollary 3.9] defined as

$r_{i}(P_{i})=[\xi_{i}(\ell_{i}(P_{i}))]^{-1}P_{i}.$

Here $\xi_{i}$ is the homeomorphism sending a $0$-symmetric ellipsoid
$C\subset\mathbb{R}^{d_{i}}$ to the unique self-adjoint positive operator
$\xi_{i}(C)\in GL(d_{i})$ such that $\xi_{i}(C)B_{2}^{d_{i}}=C,$
$i=1,\ldots,l,$ see [5, (3.4)]. We claim that the diagram below commutes and
the map $\tilde{\varphi}$ is a homeomorphism:

${\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})}$${L(d_{1})\times\cdots\times
L(d_{l})\times\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l})}$${\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})/H}$${L(d_{1})\times\cdots\times
L(d_{l})\times\left(\mathcal{E}(d_{1})\times\cdots\times\mathcal{E}(d_{l})/H\right)}$$\rho$$\varphi$$I_{\prod_{i}L(d_{i})}\times\rho_{|}$$\tilde{\varphi}$

Here $I_{\prod_{i}L(d_{i})}$ denotes the identity map of
$L(d_{1})\times\cdots\times L(d_{l})$ and $\varphi$ is defined as

$\varphi(P_{1},\ldots,P_{l})=\left(r_{1}(P_{1}),\ldots,r_{l}(P_{l}),\ell_{1}(P_{1}),\ldots,\ell_{l}(P_{l})\right),$

for $P_{i}\in\mathcal{B}(d_{i}),$ $i=1,\ldots,l.$ Observe that $\varphi$ is a
continuous map, since both $r_{i}$ and $\ell_{i}$ are continuous for each $i$.
Moreover, due to the map sending $P_{i}\in\mathcal{B}(d_{i})$ to the pair
$(r_{i}(P_{i}),\ell_{i}(P_{i}))$ is a $O(d_{i})$-homeomorphism between
$\mathcal{B}(d_{i})$ and $L(d_{i})\times\mathcal{E}(d_{i})$ (see [5, Corollary
3.9]), then $\varphi$ must be a $O(d_{1})\times\cdots\times
O(d_{l})$-equivariant homeomorphism.

On the other hand, if we let $\lambda>0$ and $X\in\mathcal{B}(d_{i}),$ then
$\lambda\xi[\ell_{i}(X)]$ is a positive self-adjoint operator such that

$\lambda\xi[\ell_{i}(X)]B_{2}^{d_{i}}=\lambda\ell_{i}(X)=\ell_{i}(\lambda X).$

Hence, from the uniqueness of $\xi[\ell_{i}(\lambda X)],$ we know that
$\xi[\ell_{i}(\lambda X)]=\lambda\xi[\ell_{i}(X)]$ and thus

(3.6) $r_{i}(\lambda X)=X,\text{ for any }\lambda>0.$

Now, if we consider arbitrary $l$-tuples $(P_{1},\ldots,P_{l})$ and
$(X_{1},\ldots,X_{l})$ in the same $H$-orbit, then there exists
$\lambda_{i}>0,$ $i=1,\ldots,l,$ such that $\lambda_{1}\cdots\lambda_{l}=1$
and $X_{i}=\lambda_{i}P_{i}.$ Therefore, by (3.6),

$\displaystyle\varphi(X_{1},\ldots,X_{l})=$
$\displaystyle\left(r_{1}(P_{1}),\ldots,r_{l}(P_{l}),\ell_{1}(\lambda_{1}P_{1}),\ldots,\ell_{l}(\lambda_{l}P_{l})\right)$
$\displaystyle=$
$\displaystyle\left(r_{1}(P_{1}),\ldots,r_{l}(P_{l}),\lambda_{1}\ell_{1}(P_{1}),\ldots,\lambda_{l}\ell_{l}(P_{l})\right),$

and so
$I_{\prod_{i}L(d_{i})}\times\rho_{|}\left(\varphi(X_{1},\ldots,X_{l})\right)=I_{\prod_{i}L(d_{i})}\times\rho_{|}\left(\varphi(P_{1},\ldots,P_{l})\right).$
Similarly, by the injectivity of $\varphi$, if the last equality holds then
$(P_{1},\ldots,P_{l})$ and $(X_{1},\ldots,X_{l})$ must be in the same
$H$-orbit. Thus, due to $I_{\prod_{i}L(d_{i})}\times\rho_{|}$ is indeed a
quotient map and $\varphi$ is a homeomorphism, this shows that
$\tilde{\varphi}$ is well-defined, bijective and it must be a homeomorphism. ∎

##  4\. On the ANR property

The support function $h_{P}:\mathbb{R}^{d}\rightarrow[0,\infty)$ of a
$0$-symmetric convex body $P\subset\mathbb{R}^{d}$ is defined as
$h_{P}(y)=\text{sup}_{x\in P}\langle x,y\rangle.$ Similarly, for a
$0$-symmetric convex body $P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}},$ its
support function
$h_{P}:\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}\rightarrow[0,\infty)$ is calculated
with respect to the Euclidean structure $\langle\cdot,\cdot\rangle_{H}$ of the
tensor space. It is well-known that $h_{P}(\cdot)$ coincides with the
Minkowski functional of the polar body. That is
$h_{P}(\cdot)=g_{P^{\circ}}(\cdot)$ ([32, Theorem 1.7.6]).

######  Proposition 4.1.

The space of tensorial bodies $\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l}),$
$d_{i}\geq 2,$ $i=1,\ldots,l$ and $l\geq 2,$ is not closed with respect to the
Minkowski sum.

###### Proof.

Let $B_{p}^{d_{i}}$ denote the closed unit ball of the classical norm
$\|\cdot\|_{p}$ on $\mathbb{R}^{d_{i}},$ $i=1,\dots,l,$ for $p=1,\infty$. We
shall show that the Minkowski sum of the projective tensor products
$P=\frac{1}{2}B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}}$ and
$R=\frac{1}{2}B_{1}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{1}^{d_{l}}$ is
not a tensorial body in $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$. Since the
support function is Minkowski additive, we know that
$h_{P+Q}(\cdot)=h_{P}(\cdot)+h_{Q}(\cdot)$ and so
$g_{(P+R)^{\circ}}(\cdot)=g_{P^{\circ}}(\cdot)+g_{R^{\circ}}(\cdot).$ Hence,
from (2.3) and the fact that $(B_{1}^{d_{i}})^{\circ}=B_{\infty}^{d_{i}}$, we
get

$g_{(P+R)^{\circ}}(x^{1}\otimes\cdots\otimes
x^{l})=\frac{\|x^{1}\|_{2}\cdots\|x^{l}\|_{2}+\|x^{1}\|_{\infty}\cdots\|x^{l}\|_{\infty}}{2},$

for any $x^{i}\in\mathbb{R}^{d_{i}},$ $i=1,\dots,l$. Thus, if we fix the
vectors $a^{i}=(1,\ldots,1)\in\mathbb{R}^{d_{i}}$ and
$b^{i}=(1,0,\ldots,0)\in\mathbb{R}^{d_{i}}$ with $i=1,\ldots,l-1,$ then, for
every $x^{l}\in\mathbb{R}^{d_{l}}$ we have

$g_{(P+R)^{\circ}}(a^{1}\otimes\cdots\otimes a^{l-1}\otimes
x^{l})=\frac{\sqrt{d_{1}\cdots d_{l-1}}\|x^{l}\|_{2}+\|x^{l}\|_{\infty}}{2},$

and

$g_{(P+R)^{\circ}}(b^{1}\otimes\cdots\otimes b^{l-1}\otimes
x^{l})=\frac{\|x^{l}\|_{2}+\|x^{l}\|_{\infty}}{2}.$

Therefore, the value of $g_{(P+R)^{\circ}}(a^{1}\otimes\cdots\otimes
a^{l-1}\otimes x^{l})/g_{(P+R)^{\circ}}(b^{1}\otimes\cdots\otimes
b^{l-1}\otimes x^{l})$ depends on $x^{l}\neq 0.$ In consequence, there do not
exist $0$-symmetric convex bodies such that (2.2) holds for $(P+R)^{\circ}.$
In particular, it is not a tensorial body. Since $P+R$ belongs to the class of
tensorial bodies if and only if $(P+R)^{\circ}$ does it too ([18, Proposition
3.5]), the proof is finished. ∎

As we just showed in the proof of the previous proposition, the Minkowski sum
of an arbitrary pair of tensorial bodies is not necessarily a tensorial body.
Nevertheless, taking into account the multilinear nature of tensors, it turns
out that in many cases the Minkowski sum does remain in the class.

######  Proposition 4.2.

Let $\lambda>0$ be a real number and let $P_{i}\in\mathcal{B}(d_{i}),$
$i=1,\dots,l,$ be an arbitrary $l$-tuple. Then, for every $j=1,\dots,l$ and
any $R_{j}\in\mathcal{B}(d_{j}),$ if
$P\in\mathcal{B}_{P_{1},\dots,P_{j},\dots,P_{l}}(d_{1},\dots,d_{l})$ and
$R\in\mathcal{B}_{P_{1},\dots,P_{j-1},R_{j},P_{j+1},\dots,P_{l}}(d_{1},\dots,d_{l}),$
then $P+\lambda R$ is a tensorial body. Indeed, $P+\lambda
R\in\mathcal{B}_{P_{1},\dots,P_{j-1},P_{j}+\lambda
R_{j},P_{j+1},\dots,P_{l}}(d_{1},\dots,d_{l}).$

###### Proof.

For simplicity of the notation, we prove the result for $j=1.$ Notice that if
$x^{1}\in P_{1},$ $y^{1}\in R_{1}$ and $x^{i}\in P_{i},$ for $i=2,\dots,l,$
then

$\displaystyle(x^{1}+\lambda y^{1})\otimes x^{2}\otimes\cdots\otimes x^{l}$
$\displaystyle=x^{1}\otimes x^{2}\otimes\cdots\otimes x^{l}+\lambda
y^{1}\otimes x^{2}\otimes\cdots\otimes x^{l}$ $\displaystyle\in
P_{1}\otimes_{\pi}P_{2}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}+\lambda
R_{1}\otimes_{\pi}P_{2}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}$
$\displaystyle\subseteq P+\lambda R.$

Thus, by the definition of $\otimes_{\pi},$ $(P_{1}+\lambda
R_{1})\otimes_{\pi}P_{2}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\subseteq
P+\lambda R.$ To prove the inclusion $P+\lambda R\subseteq(P_{1}+\lambda
R_{1})\otimes_{\epsilon}P_{2}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l},$
observe that $h_{P+\lambda R}(\cdot)=h_{P}(\cdot)+h_{\lambda R}(\cdot)$ and so
$g_{(P+\lambda R)^{\circ}}(\cdot)=g_{P^{\circ}}(\cdot)+g_{(\lambda
R)^{\circ}}(\cdot).$ Hence, by (2.3), for every $x^{i}\in\mathbb{R}^{d_{i}},$
$i=1,\dots,l,$ we also have

$\displaystyle g_{(P+\lambda R)^{\circ}}(x^{1}\otimes\cdots\otimes x^{l})$
$\displaystyle=g_{P_{1}^{\circ}}(x^{1})g_{P_{2}^{\circ}}(x^{2})\cdots
g_{P_{l}^{\circ}}(x^{l})+g_{(\lambda
R_{1})^{\circ}}(x^{1})g_{P_{2}^{\circ}}(x^{2})\cdots g_{P_{l}^{\circ}}(x^{l})$
$\displaystyle=(g_{P_{1}^{\circ}}(x^{1})+g_{(\lambda
R_{1})^{\circ}}(x^{1}))g_{P_{2}^{\circ}}(x^{2})\cdots
g_{P_{l}^{\circ}}(x^{l}).$

This, together with the definition of $\otimes_{\epsilon},$ yields to
$P+\lambda R\subseteq(P_{1}+\lambda
R_{1})\otimes_{\epsilon}P_{2}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}.$
Finally, from (2.1), it follows that $P+\lambda R$ is a tensorial body w.r.t.
$P_{1}+\lambda R_{1},P_{2},\dots,P_{l}$. This finishes the proof. ∎

As an application of Proposition 4.2, notice that it is always possible to
define a polygonal path joining any pair $P,R$ of tensorial bodies in
$\mathbb{R}^{d_{1}}\otimes\mathbb{R}^{d_{2}}.$ To this end, let us assume that
$P\in\mathcal{B}_{P_{1},P_{2}}(d_{1},d_{2})$ and
$R\in\mathcal{B}_{R_{1},R_{2}}(d_{1},d_{2}),$ for some
$P_{i},R_{i}\in\mathcal{B}(d_{i}),$ $i=1,2,$ and let $K$ be an arbitrary
tensorial body w.r.t. $P_{1},R_{2}.$ Then, the map
$\alpha:[0,1]\rightarrow\mathcal{B}_{\otimes}(d_{1},d_{2})$ defined as,

$\alpha(t)=\begin{cases}(1-2t)P+(2t)K;\text{ for }0\leq t\leq\frac{1}{2},\\\
(2-2t)K+(2t-1)R;\text{ for }\frac{1}{2}\leq t\leq 1,\end{cases}$

is a polygonal path joining $P$ and $R.$ It is not difficult to see that this
process can be extended to $\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ with
$l>2.$

######  Theorem 4.3.

$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$, $d_{i}\geq 2,$ $i=1,\dots,l,$ is
an AR. In particular, the subspace $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$
is also an AR.

###### Proof.

We shall exhibit a retraction from the space
$\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}})$ onto
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l}).$ The result then will follow from
the fact that $\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}})$ is an AR, and
that any retract of an AR is also an AR [24, §1.5].

First, for every $P\in\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}})$, let
us define the $0$-symmetric convex bodies $P^{i}\subset\mathbb{R}^{d_{i}},$
$i=1,\dots,l,$ associated to $P$ as

$P^{i}:=\left\\{x^{i}\in\mathbb{R}^{d_{i}}:\frac{e^{d_{1}}_{1}\otimes\cdots\otimes
e^{d_{i-1}}_{1}\otimes x^{i}\otimes e^{d_{i+1}}_{1}\otimes\cdots\otimes
e^{d_{l}}_{1}}{g_{P}(e^{d_{1}}_{1}\otimes e^{d_{2}}_{1}\otimes\cdots\otimes
e^{d_{l}}_{1})}\in P\right\\},$

when $i\leq l-1,$ and
$P^{l}:=\left\\{x^{l}\in\mathbb{R}^{d_{l}}:e^{d_{1}}_{1}\otimes\cdots\otimes
e^{d_{l-1}}_{1}\otimes x^{l}\in P\right\\}.$ It is not difficult to check that
the Minkowski functional associated to each $P^{i}$ is such that

$g_{P^{i}}(x^{i})=\frac{g_{P}(e^{d_{1}}_{1}\otimes\cdots\otimes
e^{d_{i-1}}_{1}\otimes x^{i}\otimes e^{d_{i+1}}_{1}\otimes\cdots\otimes
e^{d_{l}}_{1})}{g_{P}(e^{d_{1}}_{1}\otimes e^{d_{2}}_{1}\otimes\cdots\otimes
e^{d_{l}}_{1})},$

$i=1,\dots,l-1,$ and $g_{P^{l}}(x^{l})=g_{P}(e^{d_{1}}_{1}\otimes\cdots\otimes
e^{d_{l-1}}_{1}\otimes x^{l}).$ Indeed, for every $i,$ the map from
$\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}})$ to $\mathcal{B}(d_{i})$
sending $P\mapsto P^{i}$ must be continuous. To prove this, recall that for
every sequence $\\{P_{n}\\}_{n}$ and $P$ in
$\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}),$ if $P_{n}$ converges to
$P,$ then the sequence of the Minkowski functionals
$\left\\{g_{P_{n}}\right\\}_{n}$ converges uniformly to $g_{P}$ on compact
sets of $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ ([21, Lemma 3.1]). Hence, for
every $x^{i}\in\mathbb{R}^{d_{i}},$ we have that $g_{P_{n}^{i}}(x^{i})$
approaches to $g_{P^{i}}(x^{i})$. So, the sequence
$\left\\{g_{P_{n}^{i}}\right\\}_{n}$ converges pointwise to $g_{P^{i}}$ for
each $i.$ Since the pointwise convergence of Minkowski functionals implies the
uniform convergence on compact sets, $\left\\{g_{P_{n}^{i}}\right\\}_{n}$
converges uniformly to $g_{P^{i}}$ on compact sets of $\mathbb{R}^{d_{i}},$
$i=1,\ldots,l.$ In particular, each $P_{n}^{i}$ converges to $P^{i}$ with
respect to the Hausdorff metric [21, Lemma 3.1]. This proves that the map in
question is continuous.

Next, we claim that the map
$\eta:\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}})\rightarrow\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$
defined as,

$\eta(P)=\text{conv}(P\cup
P^{1}\otimes_{\pi}P^{2}\otimes_{\pi}\cdots\otimes_{\pi}P^{l})\cap
P^{1}\otimes_{\epsilon}P^{2}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P^{l},$

is the desired retraction. To prove this, notice that for every $0$-symmetric
convex body $P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}},$ $\eta(P)$ is a
$0$-symmetric convex body such that

$P^{1}\otimes_{\pi}P^{2}\otimes_{\pi}\cdots\otimes_{\pi}P^{l}\subseteq\eta(P)\subseteq
P^{1}\otimes_{\epsilon}P^{2}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P^{l},$

and thus, by (2.1), $\eta(P)$ is a tensorial body. Furthermore, from the
previous argument in combination with the continuity of both the projective
$\otimes_{\pi}$ and the injective $\otimes_{\epsilon}$ tensor products [21,
Proposition 3.3], and the fact that all the operators involved are continuous
in the class of $0$-symmetric convex bodies (see [32, Section 1.8]), we know
that $\eta$ is well-defined and continuous.

Finally, if $P$ is a tensorial body in $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}},$
then, by Remark 2.1, we know that
$P^{1}\otimes_{\pi}P^{2}\otimes_{\pi}\cdots\otimes_{\pi}P^{l}\subseteq
P\subseteq
P^{1}\otimes_{\epsilon}P^{2}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P^{l}.$
Therefore $\eta(P)=P,$ and this shows that $\eta$ is a retraction onto the
space of tensorial bodies. The last part of the statement now follows from the
fact that $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ is a retract of the space
of tensorial bodies [21, Corollary 4.10], and thus it is also an AR. ∎

As it was described in Section 2.1, for a fixed $l$-tuple
$(P_{1},\dots,P_{l})\in\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l}),$
the hyperspace $\mathcal{B}_{P_{1},\dots,P_{l}}(d_{1},\dots,d_{l})$ consists
of the tensorial bodies $P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ such
that

$P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\subseteq P\subseteq
P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}.$

In [18, Proposition 3.1], it is proved that this hyperspace coincides with the
set of closed unit balls of reasonable crossnorms on the space
$\otimes_{i=1}^{l}M_{i},$ with $M_{i}=(\mathbb{R}^{d_{i}},g_{P_{i}})$. In this
case, the map sending $P$ to its Minkowski functional $g_{P}$ gives us a
bijective correspondence between
$\mathcal{B}_{P_{1},\dots,P_{l}}(d_{1},\dots,d_{l})$ and the set of reasonable
crossnorms on $\otimes_{i=1}^{l}M_{i}.$ Since, each tensorial body
$P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ belongs to
$\mathcal{B}_{P_{1},\dots,P_{l}}(d_{1},\dots,d_{l}),$ for some
$P_{i}\in\mathcal{B}(d_{i}),$ $i=1,\dots,l,$ the following relation holds:

$\mathcal{B}_{\otimes}(d_{1},\dots.,d_{l})=\bigcup_{(P_{1},\dots,P_{l})\in\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})}\mathcal{B}_{P_{1},\dots,P_{l}}(d_{1},\dots,d_{l}).$

It is noteworthy that, in contrast with
$\mathcal{B}_{\otimes}(d_{1},\dots.,d_{l})$, each
$\mathcal{B}_{P_{1},\dots,P_{l}}(d_{1},\dots,d_{l})$ is a compact and convex
subspace of $\mathcal{B}(\otimes_{i=1}^{l}\mathbb{R}^{d_{i}})$ [21,
Proposition 3.4]. These properties allow us to determine its topological
structure.

######  Proposition 4.4.

Let $d_{i}\geq 2$, $i=1,\ldots,l.$ Then, for every tuple of $0$-symmetric
convex bodies $P_{i}\subset\mathbb{R}^{d_{i}}$, the hyperspace
$\mathcal{B}_{P_{1},\ldots,P_{l}}(d_{1},\ldots,d_{l})$ is homeomorphic to
$\mathcal{Q}.$

###### Proof.

The proof is a direct consequence of Lemma 2.6, and the fact that
$P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\subsetneqq
P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}$ for every $l$-tuple
$P_{i}\in\mathcal{B}(d_{i}),$ $i=1,\dots,l,$ [7, Theorem 2]. ∎

##  5\. Topology of the slice $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$

In this section, we achieve our main goal. Namely, we show that
$\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})$ is homeomorphic to the Hilbert
cube $\mathcal{Q}$ (Theorem 5.4) and, in consequence, the space of tensorial
bodies $\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ is homeomorphic to
$\mathcal{Q}\times\mathbb{R}^{p}$ (Corollary 5.5).

Recall that $\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})$ consists of the
tensorial bodies $P\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ for which
$\ell_{\otimes}(P)=B_{2}^{d_{1},\ldots,d_{l}}.$ It is known that
$\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})$ is a compact $O_{\otimes}$-global
slice for the $GL_{\otimes}$-space $\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$
(see Theorem 2.3 and [21, Sections 4.1 and 4.2]). For simplicity, we denote by
$\Pi\cap\mathcal{L}_{\otimes}$ the subset
$\Pi(d_{1},\ldots,d_{l})\cap\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})$
consisting of the tensor products
$P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l},$ $P_{i}\in\mathcal{B}(d_{i}),$
$i=1,\dots,l,$ such that
$\ell_{\otimes}(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l})=B_{2}^{d_{1},\ldots,d_{l}}.$
We follow the notation of [1, Remark 1] (c.f. [5, Section 3.2]), where the
class of $0$-symmetric convex bodies $P\subset\mathbb{R}^{d}$ such that
$Low(P)=B_{2}^{d}$ is denoted by $L(d).$

######  Proposition 5.1.

The following hold:

  1. (1)

$\Pi\cap\mathcal{L}_{\otimes}$ is an $O_{\otimes}$-strong deformation retract
of $\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l}).$

  2. (2)

The map $\otimes_{\pi}$ restricted to $L(d_{1})\times\cdots\times L(d_{l})$ is
a homeomorphism onto $\Pi\cap\mathcal{L}_{\otimes}.$ In particular,
$\Pi\cap\mathcal{L}_{\otimes}$ is homeomorphic to $\mathcal{Q}$.

  3. (3)

$\Pi\cap\mathcal{L}_{\otimes}$ is $O_{\otimes}$-strictly contractible to the
$O_{\otimes}$-fixed point
$B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}}.$

###### Proof.

1\. Let us denote by $conv_{\otimes|\mathcal{L}_{\otimes}}$ the restriction of
the $GL_{\otimes}$-map $conv_{\otimes}$ to
$\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l}).$ Below we prove that
$conv_{\otimes|\mathcal{L}_{\otimes}}$ is an $O_{\otimes}$-strong deformation
retraction. To this end, observe that $conv_{\otimes|\mathcal{L}_{\otimes}}$
maps $\mathcal{L}_{\otimes}$ onto $\Pi\cap\mathcal{L}_{\otimes},$ due to the
map $conv_{\otimes}$ being a retraction of $\mathcal{B}_{\otimes}$ onto $\Pi$
[21, Proposition 4.4] and
$\ell_{\otimes}(conv_{\otimes}(P))=\ell_{\otimes}(P)=B_{2}^{d_{1},\ldots,d_{l}},$
for every $P\in\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l}).$

Now let us define
$H:\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})\times[0,1]\rightarrow\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})$
as

$H(P,t)=(1-t)P+tconv_{\otimes}(P).$

We will prove that $H$ is the desired $O_{\otimes}$-homotopy. Notice that $H$
is well-defined, since
$conv_{\otimes}(P)=P^{1}\otimes_{\pi}\cdots\otimes_{\pi}P^{1}$, where
$P^{1},\dots,P^{l}$ are the convex bodies associated to $P$ in Remark 2.1, see
[21, p. 12]. In consequence, both $P$ and $conv_{\otimes}(P)$ belong to
$\mathcal{B}_{P^{1},\dots,P^{l}}(d_{1},\dots,d_{1}),$ which is convex [21,
Proposition 3.4]. Thus, $H(P,t)$ is a tensorial body such that
$\ell_{\otimes}((1-t)P+tconv_{\otimes}(P))=\ell_{\otimes}(P)=B_{2}^{d_{1},\ldots,d_{l}}.$
The $O_{\otimes}$-equivariance and continuity of $H$ follow directly from
those of the map $conv_{\otimes}$ and the properties of the Minkowski sum.
Clearly, $H(P,0)=P$ and $H(P,1)=conv_{\otimes}(P)$ for every
$P\in\mathcal{L}_{\otimes}(d_{1},\dots,d_{l}).$ Furthermore, if
$P\in\Pi\cap\mathcal{L}_{\otimes},$ then $conv_{\otimes}(P)=P$ and $H(P,t)=P$
for any $t\in[0,1].$ Thus, $H$ is an $O_{\otimes}$-strong homotopy as
required.

2\. Let us first prove that $\otimes_{\pi}$ restricted to
$L(d_{1})\times\cdots\times L(d_{l})$ is a homeomorphism onto its image
$\otimes_{\pi}(L(d_{1}),\ldots,L(d_{l})).$ To this end, notice that
$\otimes_{\pi|L(d_{1})\times\cdots\times L(d_{l})}$ is surjective and, since
$\otimes_{\pi}$ is continuous and $L(d_{1})\times\cdots\times L(d_{l})$ is
compact ([5, Proposition 3.4]), $\otimes_{\pi|L(d_{1})\times\cdots\times
L(d_{l})}$ must be a closed map. To prove that it is injective, suppose that
$R_{1}\otimes_{\pi}\cdots\otimes_{\pi}R_{l}=P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l},$
with $R_{i},P_{i}\in L(d_{i}),$ then, by [18, Proposition 3.6], there exist
$\lambda_{i}>0,$ $i=1,\ldots,l,$ such that $\lambda_{1}\cdots\lambda_{l}=1$
and $P_{i}=\lambda_{i}R_{i}.$ However, each pair $R_{i},P_{i}\in L(d_{i}),$
hence
$B_{2}^{d_{i}}=Low(P_{i})=\lambda_{i}Low(R_{i})=\lambda_{i}B_{2}^{d_{i}},$ and
so $\lambda_{i}=1.$ This shows that $P_{i}=R_{i}$ and therefore
$\otimes_{\pi|L(d_{1})\times\cdots\times L(d_{l})}$ is a homeomorphism.

Now we will prove that
$\otimes_{\pi}(L(d_{1}),\ldots,L(d_{l}))=\Pi\cap\mathcal{L}_{\otimes}.$ Let
$R_{i}\in L(d_{i})$ for $i=1,\ldots,l.$ Since the Löwner ellipsoid of
$R_{1}\otimes_{\pi}\cdots\otimes_{\pi}R_{l}$ is the Hilbertian tensor product
$Low(R_{1})\otimes_{2}\cdots\otimes_{2}Low(R_{l})$ (see [8, Lemma 1]), we have

$\ell_{\otimes}(R_{1}\otimes_{\pi}\cdots\otimes_{\pi}R_{l})=Low(R_{1})\otimes_{2}\cdots\otimes_{2}Low(R_{l})=B_{2}^{d_{1}}\otimes_{2}\cdots\otimes_{2}B_{2}^{d_{l}}=B_{2}^{d_{1},\ldots,d_{l}}.$

Hence $\otimes_{\pi}(L(d_{1}),\ldots,L(d_{l}))$ is contained in
$\Pi\cap\mathcal{L}_{\otimes}.$ On the other hand, if
$R\in\Pi\cap\mathcal{L}_{\otimes},$ then
$R=conv_{\otimes}(R)=R^{1}\otimes_{\pi}\cdots\otimes_{\pi}R^{l}$ (see [21, p.
12]) and

$\ell_{\otimes}(R)=Low(R^{1})\otimes_{2}\cdots\otimes_{2}Low(R^{l})=B_{2}^{d_{1}}\otimes_{2}\cdots\otimes_{2}B_{2}^{d_{l}}.$

Thus, by [18, Proposition 3.6], there exist $\lambda_{i}>0,$ $i=1,\ldots,l,$
such that $\lambda_{1}\cdots\lambda_{l}=1$ and
$Low(R^{i})=\lambda_{i}B_{2}^{d_{i}}.$ Therefore,
$P_{i}:=\lambda_{i}^{-1}R^{i}$ belongs to $L(d_{i})$ and, from [19,
Proposition 5], we have

$\displaystyle
R=R^{1}\otimes_{\pi}\cdots\otimes_{\pi}R^{l}=\lambda_{1}P_{1}\otimes_{\pi}\cdots\otimes_{\pi}\lambda_{l}P_{l}=(\lambda_{1}\cdots\lambda_{l})P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}=P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}.$

So, $R\in\otimes_{\pi}(L(d_{1}),\ldots,L(d_{l})).$ This proves the reverse
inclusion. The result now follows from two facts: Each $L(d_{i}),$
$i=1,\ldots,l,$ is homemorphic to the Hilbert cube $\mathcal{Q}$ [1], and
$\mathcal{Q}^{k},$ $k\in\mathbb{N},$ is homeomorphic to $\mathcal{Q}$. (see
[24, Excercise 2, §1.1]).

3\. Let us denote by $\rho_{i}$, $i=1,\ldots,l,$ the canonical projection of
$L(d_{1})\times\cdots\times L(d_{l})$ onto $L(d_{i})$. Also, observe as a
consequence of 1., the maps defined by
$f_{i}(P)=\rho_{i}\circ(\otimes_{\pi|L(d_{1})\times\cdots\times
L(d_{l})})^{-1}(P)$, $P\in\Pi\cap\mathcal{L}_{\otimes},$ are continuous on
$\Pi\cap\mathcal{L}_{\otimes}$ and

(5.1) $f_{1}(P)\otimes_{\pi}\cdots\otimes_{\pi}f_{l}(P)=P.$

We will prove that the map
$F:\Pi\cap\mathcal{L}_{\otimes}\times[0,1]\rightarrow\Pi\cap\mathcal{L}_{\otimes},$
defined as

$F(P,t)=((1-t)f_{1}(P)+tB_{2}^{d_{1}})\otimes_{\pi}\cdots\otimes_{\pi}((1-t)f_{l}(P)+tB_{2}^{d_{l}}),$

is the desired $O_{\otimes}$-strict homotopy. Since
$Low((1-t)f_{i}(P)+tB_{2}^{d_{i}})=B_{2}^{d_{i}}$ for each $i,$ then
$F(P,t)\in\Pi\cap\mathcal{L}_{\otimes}$ and $F$ is well-defined. Its
continuity is straightforward from the fact $\otimes_{\pi}$ and $f_{i}$ are
continuous. Clearly
$F(P,0)=f_{1}(P)\otimes_{\pi}\cdots\otimes_{\pi}f_{l}(P)=P$ and
$F(P,1)=B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}},$ for every
$P$.

To prove that $F_{t}$ is an $O_{\otimes}$-map for any $t\in[0,1]$, let $U$ be
an orthogonal map in $O_{\otimes}(d_{1},\dots,d_{l})$ and
$P\in\Pi\cap\mathcal{L}_{\otimes}.$ By [21, Lemma A.1],
$U=U_{1}\otimes\cdots\otimes U_{l}\circ U_{\sigma},$ for some $U_{i}\in
O(d_{i}),$ $i=1,\dots,l,$ and $U_{\sigma}\in\mathcal{P}.$ Hence, by the
identity (5.1), $UP=U(f_{1}(P)\otimes_{\pi}\cdots\otimes_{\pi}f_{l}(P)),$ for
every $P\in\Pi\cap\mathcal{L}_{\otimes}$ and

$\displaystyle F_{t}(UP)=$ $\displaystyle
F_{t}(U(f_{1}(P)\otimes_{\pi}\cdots\otimes_{\pi}f_{l}(P)))$ $\displaystyle=$
$\displaystyle
F_{t}(U_{1}f_{\sigma(1)}(P)\otimes_{\pi}\cdots\otimes_{\pi}U_{l}f_{\sigma(l)}(P))$
$\displaystyle=$
$\displaystyle((1-t)U_{1}f_{\sigma(1)}(P)+tB_{2}^{d_{1}})\otimes_{\pi}\cdots\otimes_{\pi}((1-t)U_{l}f_{\sigma(l)}(P)+tB_{2}^{d_{l}})$
$\displaystyle=$ $\displaystyle
U_{1}((1-t)f_{\sigma(1)}(P)+tB_{2}^{d_{1}})\otimes_{\pi}\cdots\otimes_{\pi}U_{l}((1-t)f_{\sigma(l)}(P)+tB_{2}^{d_{l}})$
$\displaystyle\overset{*}{=}$ $\displaystyle U_{1}\otimes\cdots\otimes
U_{l}U_{\sigma}((1-t)f_{1}(P)+tB_{2}^{d_{1}})\otimes_{\pi}\cdots\otimes_{\pi}((1-t)f_{l}(P)+tB_{2}^{d_{1}})$
$\displaystyle=$ $\displaystyle UF_{t}(P).$

(*) Follows from the identity (2.5). Finally, for $t\in(0,1),$
$F(P,t)=B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}}$ only if
there is $\lambda_{i}>0,$ $i=1,\ldots,l,$ such that
$\lambda_{1}\cdots\lambda_{l}=1$ and
$(1-t)f_{i}(P)+tB_{2}^{d_{i}}=\lambda_{i}B_{2}^{d_{i}}$ [18, Proposition 3.6].
Then, $Low((1-t)f_{i}(P)+tB_{2}^{d_{i}})=Low(\lambda_{i}B_{2}^{d_{i}})$ and
$B_{2}^{d_{i}}=\lambda_{i}B_{2}^{d_{i}}.$ Thus each $\lambda_{i}=1$ and, due
to $t\in(0,1),$ $f_{i}(P)=B_{2}^{d_{i}}$ for $i=1,\dots,l.$ Therefore,
$P=B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}}.$ This proves
that $F$ is a strict $O_{\otimes}$-homotopy. ∎

In Proposition 5.1 above, we have showed that
$\otimes_{\pi|L(d_{1})\times\cdots\times L(d_{l})}$ is a homeomorphism onto
$\Pi\cap\mathcal{L}_{\otimes}.$ In this way, since for each
$P\in\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l}),$
$conv_{\otimes}(P)\in\Pi\cap\mathcal{L}_{\otimes},$ we know that there is only
one $l$-tuple $(\bar{P}_{1},\ldots,\bar{P}_{l})$ in
$L(d_{1})\times\cdots\times L(d_{l})$ such that
$conv_{\otimes}(P)=\bar{P}_{1}\otimes_{\pi}\cdots\otimes_{\pi}\bar{P}_{l}.$

A map $f:A\rightarrow A,$ defined on a metric space $(A,d_{A}),$ will be
called $\varepsilon$-close to the identity of $A$ if and only if
$d_{A}(f(x),x)<\varepsilon,$ for every $x\in A$.

######  Lemma 5.2.

For each $i=1,\ldots,l,$ let $\mathcal{A}_{i}$ be a non-empty subset of
$L(d_{i}),$ and denote by $\mathcal{A}$ the set
$(\text{conv}_{\otimes|\mathcal{L}_{\otimes}})^{-1}(\otimes_{\pi}(\mathcal{A}_{1},\ldots,\mathcal{A}_{l}))\subseteq\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})$.
For every $\rho\in(0,\infty),$ let $\mathcal{F}_{\rho}$ be the family of all
$l$-tuples $(f^{1},\ldots,f^{l})$ of continuous maps
$f^{i}:\mathcal{A}_{i}\rightarrow\mathcal{A}_{i}$ such that each $f_{i}$ is
$\rho$-close to the identity of $\mathcal{A}_{i}$. Then for every
$\varepsilon>0,$ there exists $\delta>0$ such that for each
$(f^{1},\ldots,f^{l})\in\mathcal{F}_{\delta},$ the map
$\tilde{f}_{\varepsilon}:\mathcal{A}\rightarrow\mathcal{A}$ defined as,

(5.2) $\tilde{f}_{\varepsilon}(P)=\text{conv}\left(P\cup
f^{1}(\bar{P}_{1})\otimes_{\pi}\cdots\otimes_{\pi}f^{l}(\bar{P}_{l})\right)\cap
f^{1}(\bar{P}_{1})\otimes_{\epsilon}\cdots\otimes_{\epsilon}f^{l}(\bar{P}_{l}),$

is a continuous map $\varepsilon$-close to the identity of $\mathcal{A}.$

###### Proof.

To shorten notation, let us simply write $\mathcal{L}_{\otimes}.$ Clearly,
$\mathcal{A}$ is a non-empty subset of $\mathcal{L}_{\otimes}.$ Since
$\mathcal{L}_{\otimes}$ is compact [21, Proposition 4.6], the operations
convex hull, union and intersection are uniformly continuous on it [32,
Section 1.8]. For this reason, given $\varepsilon>0,$ there exists $\eta>0$
(only depending on $\varepsilon$) such that if $X,X^{\prime},Y,Y^{\prime},Z$
and $Z^{\prime}$ belong to $\mathcal{L}_{\otimes}$ and satisfy
$\delta^{H}(X,X^{\prime})$, $\delta^{H}(Y,Y^{\prime})$,
$\delta^{H}(Z,Z^{\prime})<\eta,$ then

(5.3) $\delta^{H}\left(\text{conv}(X\cup Y)\cap Z,\text{conv}(X^{\prime}\cup
Y^{\prime})\cap Z^{\prime}\right)<\varepsilon.$

On the other hand, from the compactness of each $L(d_{i})$ [5, Proposition
3.4], it follows that the maps $\otimes_{\pi}$ and $\otimes_{\epsilon}$ are
uniformly continuous on $L(d_{1})\times\cdots\times L(d_{l})$ [21, Proposition
3.3]. In consequence, for the above mentioned $\eta,$ there exists $\delta>0$
such that if $\delta^{H}(P_{i},R_{i})<\delta,$ $i=1,\ldots,l,$ then

(5.4)
$\delta^{H}(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l},R_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}R_{l})<\eta,\quad\alpha=\pi,\epsilon,$

for every pair of tuples $(P_{1},\ldots,P_{l})$ and $(R_{1},\ldots,R_{l})$ in
$L(d_{1})\times\cdots\times L(d_{l}).$ Observe that $\delta$ only depends of
$\varepsilon.$ Next, let $(f^{1},\ldots,f^{l})$ be an $l$-tuple in
$\mathcal{F}_{\delta}$. Then for every $P\in\mathcal{A},$
$\delta^{H}(f^{i}(\bar{P}_{i}),\bar{P}_{i})<\delta,$ $i=1,\dots,l.$ Hence, by
(5.4),
$\delta^{H}(f^{1}(\bar{P}_{1})\otimes_{\alpha}\cdots\otimes_{\alpha}f^{l}(\bar{P}_{l}),\bar{P}_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}\bar{P}_{l})<\eta,$
$\alpha=\pi,\epsilon.$ Moreover, since
$P\in\mathcal{B}_{\bar{P}_{1},\ldots,\bar{P}_{l}}(d_{1},\dots,d_{l})$, we have
that

(5.5)
$P=\text{conv}(P\cup\bar{P}_{1}\otimes_{\pi}\cdots\otimes_{\pi}\bar{P}_{l})\cap\bar{P}_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}\bar{P}_{l},$

for each $P\in\mathcal{A}\subseteq\mathcal{L}_{\otimes}.$ Thus,

$\displaystyle\delta^{H}$
$\displaystyle\left(\text{conv}(P\cup\bar{P}_{1}\otimes_{\pi}\cdots\otimes_{\pi}\bar{P}_{l}),\text{conv}(P\cup
f^{1}(\bar{P}_{1})\otimes_{\pi}\cdots\otimes_{\pi}f^{l}(\bar{P}_{l}))\right)$
$\displaystyle\leq\delta^{H}\left(\bar{P}_{1}\otimes_{\pi}\cdots\otimes_{\pi}\bar{P}_{l},f^{1}(\bar{P}_{1})\otimes_{\pi}\cdots\otimes_{\pi}f^{l}(\bar{P}_{l})\right)<\eta.$

Therefore, from (5.3), the identity in (5.5) and the last inequality, it
follows that the map $\tilde{f}_{\varepsilon}$ associated to the tuple
$(f^{1},\ldots,f^{l})$ satisfies that
$\delta^{H}(\tilde{f}_{\varepsilon}(P),P)<\varepsilon.$ We claim that
$\tilde{f}_{\varepsilon}(P)\in\mathcal{A},$ for each $P\in\mathcal{A}.$
Indeed, if $P\in\mathcal{A},$ notice that
$f^{i}(\bar{P}_{i})\in\mathcal{A}_{i},$ $i=1,\dots,l,$ and
$\tilde{f}_{\varepsilon}(P)\in\mathcal{B}_{f^{1}(\bar{P}_{1}),\ldots,f^{l}(\bar{P}_{l})}(d_{1},\dots,d_{l})$.
Then, by [21, (4.1)],
$conv_{\otimes}(\tilde{f}_{\varepsilon}(P))=f^{1}(\bar{P}_{1})\otimes_{\pi}\cdots\otimes_{\pi}f^{l}(\bar{P}_{l})$
belongs to $\otimes_{\pi}(\mathcal{A}_{1},\ldots,\mathcal{A}_{l}).$ This shows
that $\tilde{f}_{\varepsilon}$ is a map $\varepsilon$-close to the identity of
$\mathcal{A}.$

It remains to prove that
$\tilde{f}_{\varepsilon}:\mathcal{A}\rightarrow\mathcal{A}$ is continuous.
Indeed, this follows directly from the continuity of the tensor products
$\otimes_{\pi}$, $\otimes_{\epsilon}$ on
$\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{1})$ [21, Proposition
3.3], and the continuity of the operations convex hull, union and intersection
on the hyperspace of $0$-symmetric convex bodies. ∎

In [2, Lemma 5.6], it is proved that $\\{B_{2}^{d}\\},$ $d\geq 2,$ is a
$Z$-set in $L(d).$ To achieve this, a family of maps
$\mathcal{X}_{\rho}:L(d)\rightarrow L_{0}(d),$ $\rho>0,$ $\rho$-close to the
identity of $L(d)$ is constructed. Here
$L_{0}(d):=L(d)\setminus\\{B_{2}^{d}\\}.$ In the next proposition, we use
Lemma 5.2 to prove that

$\mathcal{L}_{\otimes}^{*}(d_{1},\ldots,d_{l}):=\left\\{P\in\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l}):\bar{P}_{i}\in
L_{0}(d_{i}),i=1,\ldots,l\right\\}$

is the complement of a $Z$-set in $\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l}).$

######  Proposition 5.3.

For any $\varepsilon>0,$ there exists a continuous map
$\tilde{\mathcal{X}}_{\varepsilon}:\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})\\\
\rightarrow\mathcal{L}_{\otimes}^{*}(d_{1},\ldots,d_{l})$ $\varepsilon$-close
to the identity of $\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l}).$ In fact,
$\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})\setminus\mathcal{L}_{\otimes}^{*}(d_{1},\ldots,d_{l})$
is a $Z$-set in $\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l}).$

###### Proof.

To shorten notation we simply write $\mathcal{L}_{\otimes}^{*}.$ First, notice
that $\mathcal{L}_{\otimes}^{*}$ satisfies the following:

(5.6)
$\mathcal{L}_{\otimes}^{*}=(\text{conv}_{\otimes|\mathcal{L}_{\otimes}})^{-1}(\otimes_{\pi}\left(L_{0}(d_{1}),\ldots
L_{0}(d_{l}))\right).$

This is straightforward from the definition of the $l$-tuple $\bar{P}_{i},$
$i=1,\ldots,l,$ associated to each $P\in\mathcal{L}_{\otimes}$. Furthermore,
$\mathcal{L}_{\otimes}^{*}$ is open due to the fact that
$\otimes_{\pi|L(d_{1})\times\cdots\times L(d_{l})}$ is a homeomorphism
(Proposition 5.1), $\text{conv}_{\otimes|\mathcal{L}_{\otimes}}$ is continuous
and each $L_{0}(d_{i})$ is open in $L(d_{i})$. As a consequence, its
complement $\mathcal{L}_{\otimes}\setminus\mathcal{L}_{\otimes}^{*}$ is closed
in $\mathcal{L}_{\otimes}.$

Now, if we consider the family of continuous maps $\rho$-close to the identity
of $L(d_{i})$ constructed in [2, Lemma 5.6],
$\mathcal{X}^{i}_{\rho}:L(d_{i})\rightarrow L_{0}(d_{i}),$ $\rho>0$ and
$i=1,\ldots,l$. Then, from Lemma 5.2, we know that, for every $\varepsilon>0,$
there exists $\delta>0$ such that the map
$\tilde{\mathcal{X}}_{\varepsilon}:\mathcal{L}_{\otimes}\rightarrow\mathcal{L}_{\otimes},$
defined as

$\tilde{\mathcal{X}}_{\varepsilon}(P)=\text{conv}(P\cup\mathcal{X}_{\delta}^{1}(\bar{P}_{1})\otimes_{\pi}\cdots\otimes_{\pi}\mathcal{X}_{\delta}^{l}(\bar{P}_{l}))\cap\mathcal{X}_{\delta}^{1}(\bar{P}_{1})\otimes_{\epsilon}\cdots\otimes_{\epsilon}\mathcal{X}_{\delta}^{l}(\bar{P}_{l}),$

is continuous and $\varepsilon$-close to the identity of
$\mathcal{L}_{\otimes}.$ We have used that

$\mathcal{L}_{\otimes}=(\text{conv}_{\otimes|\mathcal{L}_{\otimes}})^{-1}(\otimes_{\pi}\left(L(d_{1}),\ldots
L(d_{l}))\right).$

The proof is completed by showing that
$\tilde{\mathcal{X}}_{\varepsilon}(P)\in\mathcal{L}_{\otimes}^{*},$ for any
$P\in\mathcal{L}_{\otimes}.$ This follows from the fact that
$\tilde{\mathcal{X}}_{\varepsilon}(P)$ is a tensorial body with respect to
$\mathcal{X}_{\delta}^{1}(\bar{P}_{1}),\ldots,\mathcal{X}_{\delta}^{l}(\bar{P}_{l}),$
and so
$conv_{\otimes}(\tilde{\mathcal{X}}_{\varepsilon}(P))=\mathcal{X}_{\delta}^{1}(\bar{P}_{1})\otimes_{\pi}\cdots\otimes_{\pi}\mathcal{X}_{\delta}^{l}(\bar{P}_{l})$
(see [21, (4.1)]). ∎

In the proof of the theorem below, by $\mathcal{P}(d),$ $d\geq 2,$ we denote
the subset of $L_{0}(d)$ consisting of the $P\in L_{0}(d)$ such that the
contact set $\partial P\cap\partial B_{2}^{d}$ has empty interior relative to
$\partial B_{2}^{d}.$ $\mathcal{P}(d)$ was introduced in [2].

######  Theorem 5.4.

Let $d_{i}\geq 2,$ $i=1,\dots,l$. Then
$\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ is homeomorphic to the Hilbert
cube.

###### Proof.

Recall that $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ is a compact
contractible AR ([21, Proposition 4.6] and Theorem 4.3), then, by the
characterization of $\mathcal{Q}$ as the only $\mathcal{Q}$-manifold which is
both compact and contractible [24, Theorem 7.5.8], we only need to show that
$\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ is a $\mathcal{Q}$-manifold. For
this purpose, we will prove that
$\mathcal{L}_{\otimes}^{*}(d_{1},\ldots,d_{l})$ is an open
$\mathcal{Q}$-manifold in $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l}),$ whose
complement is a $Z$-set (see Proposition 5.3). Then, by [35, §3], we will have
that $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ is a $\mathcal{Q}$-manifold.

We begin by proving that $\mathcal{L}_{\otimes}^{*}$ is a locally compact ANR.
To this end, notice that in the proof of Proposition 5.3, we showed that
$\mathcal{L}_{\otimes}^{*}$ is open in $\mathcal{L}_{\otimes}.$ Hence, given
that $\mathcal{L}_{\otimes}$ is a compact AR (Theorem 4.3),
$\mathcal{L}_{\otimes}^{*}$ must be a locally compact ANR.

To prove that $\mathcal{L}_{\otimes}^{*}$ is a Hilbert cube manifold, we use
Toruńczyk’s characterization [35, Theorem 1]. In this sense, it is enough to
show that for each $\varepsilon>0,$ there are two continuous maps
$\tilde{f}_{\varepsilon},\tilde{h}_{\varepsilon}:\mathcal{L}_{\otimes}^{*}\rightarrow\mathcal{L}_{\otimes}^{*}$
$\varepsilon$-close to the identity of $\mathcal{L}_{\otimes}^{*}$ such that
$\text{Im}(\tilde{f}_{\varepsilon})\cap\text{Im}(\tilde{h}_{\varepsilon})=\emptyset.$
Let $f_{\rho}^{i}:L_{0}(d_{i})\rightarrow\mathcal{P}(d_{i})$ and
$h_{\rho}^{i}:L_{0}(d_{i})\rightarrow
L_{0}(d_{i})\setminus\mathcal{P}(d_{i}),$ $\rho>0$, be the families of
continuous maps, defined in [2, Lemma 5.4], $\rho$-close to the identity of
$L_{0}(d_{i}),$ $i=1,\dots,l.$ Observe that, as a consequence of (5.6), if we
let $\mathcal{A}_{i}=L_{0}(d_{i})$ in Lemma 5.2, then for every
$\varepsilon>0,$ there exists $\delta>0$ such that the maps
$\tilde{f}_{\varepsilon},\tilde{h}_{\varepsilon}:\mathcal{L}_{\otimes}^{*}\rightarrow\mathcal{L}_{\otimes}^{*}$
defined as

$\tilde{f}_{\varepsilon}(P)=\text{conv}(P\cup
f_{\delta}^{1}(\bar{P}_{1})\otimes_{\pi}\cdots\otimes_{\pi}f_{\delta}^{l}(\bar{P}_{l}))\cap
f_{\delta}^{1}(\bar{P}_{1})\otimes_{\epsilon}\cdots\otimes_{\epsilon}f_{\delta}^{l}(\bar{P}_{l}),$

and

$\tilde{h}_{\varepsilon}(P)=\text{conv}(P\cup
h_{\delta}^{1}(\bar{P}_{1})\otimes_{\pi}\cdots\otimes_{\pi}h_{\delta}^{l}(\bar{P}_{l}))\cap
h_{\delta}^{1}(\bar{P}_{1})\otimes_{\epsilon}\cdots\otimes_{\epsilon}h_{\delta}^{l}(\bar{P}_{l})$

are continuous and $\varepsilon$-close to the identity of
$\mathcal{L}_{\otimes}^{*}.$ Moreover, by [21, (4.1)], due to
$\tilde{f}_{\varepsilon}(P)$ and $\tilde{h}_{\varepsilon}(P)$ are tensorial
bodies with respect to
$f_{\delta}^{1}(\bar{P}_{1}),\dots,f_{\delta}^{l}(\bar{P}_{l})$ and
$h_{\delta}^{1}(\bar{P}_{1}),\dots,h_{\delta}^{l}(\bar{P}_{l}),$ respectively,
we have that

$conv_{\otimes}(\tilde{f}_{\varepsilon}(P))=f_{\delta}^{1}(\bar{P}_{1})\otimes_{\pi}\cdots\otimes_{\pi}f_{\delta}^{l}(\bar{P}_{l})\in\otimes_{\pi}(\text{Im}(f_{\delta}^{1})\times\cdots\times\text{Im}(f_{\delta}^{l})),$

and
$conv_{\otimes}(\tilde{h}_{\varepsilon}(P))=h_{\delta}^{1}(\bar{P}_{1})\otimes_{\pi}\cdots\otimes_{\pi}h_{\delta}^{l}(\bar{P}_{l})\in\otimes_{\pi}(\text{Im}(h_{\delta}^{1})\times\cdots\times\text{Im}(h_{\delta}^{l})),$
for any $P\in\mathcal{L}_{\otimes}^{*}.$ Hence, since
$\otimes_{\pi|L(d_{1})\times\cdots\times L(d_{l})}$ is a homeomorphism
(Proposition 5.1), $\text{Im}(f_{\delta}^{i})\in\mathcal{P}(d_{i})$ and
$\text{Im}(h_{\delta}^{i})\in L_{0}(d_{i})\setminus\mathcal{P}(d_{i}),$
$i=1,\dots,l,$ then the images $\tilde{f}_{\varepsilon}$ and
$\tilde{h}_{\varepsilon}$ must be disjoint. ∎

######  Corollary 5.5.

Let $d_{i}\geq 2,$ $i=1,\ldots,l$ and
$p=\frac{d_{1}(d_{1}+1)+\cdots+d_{l}(d_{l}+1)}{2}.$ Then, the space of
tensorial bodies $\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ is homeomorphic
to $\mathcal{Q}\times\mathbb{R}^{p}.$ Particularly,
$\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ and
$\Pi_{\otimes}(d_{1},\ldots,d_{l})$ are homeomorphic.

###### Proof.

It is a direct consequence of Theorem 5.4 and [21, Corollary 4.11]. In the
latter, it is proved that $\mathcal{B}_{\otimes}(d_{1},\ldots,d_{l})$ is
homeomorphic to
$\mathcal{L}_{\otimes}(d_{1},\ldots,d_{l})\times\mathbb{R}^{p}$, with $p$ as
in the statement of the corollary. The last part of the statement follows
directly from Theorem 3.2-3. ∎

##  6\. Final remarks

Let us finish the paper by exploring the relation between the so called
Banach-Mazur compactum and the orbit space
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$. To
this end, we first introduce some notation. The Banach-Mazur compactum is
denoted by $\mathcal{BM}(d).$ It is the set of equivalence classes of
$0$-symmetric convex bodies in $\mathbb{R}^{d}$ determined by the relation:
$P\sim R$ if and only if $\delta^{BM}(P,R)=1.$ Here,
$\delta^{BM}(\cdot,\cdot)$ is the so called Banach-Mazur distance [20, p.
309]. It is well-known that $\mathcal{BM}(d)$ is a compact metric space with
the metric induced by the logarithm $\text{log }\delta_{BM}.$ In terms of
topological groups, another well-known fact is that the Banach-Mazur compactum
$(\mathcal{BM}(d),\text{log }\delta^{BM})$ is homeomorphic to the orbit space
$\mathcal{B}(d)/GL(d)$ (see e.g. [26, p. 1191]) or [31, Section 6] and the
references therein). In a similar way, for the class of tensorial bodies a
Banach-Mazur type compactum can be defined. In this case, the compactum of
tensorial bodies $\mathcal{BM}_{\otimes}(d_{1},\dots,d_{l})$ is defined as the
set of equivalence classes of tensorial bodies in
$\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ determined by the relation:
$P\sim^{\prime}R$ if and only if $\delta_{\otimes}^{BM}(P,R)=1.$ Here,
$\delta_{\otimes}^{BM}(\cdot,\cdot)$ denotes the tensorial Banach-Mazur
distance (2.7). In [21], it is proved that $\text{log }\delta_{\otimes}^{BM}$
is a metric on $\mathcal{BM}_{\otimes}(d_{1},\dots,d_{l})$ which turns it into
a compact metric space. Indeed, a fundamental property is that
$(\mathcal{BM}_{\otimes}(d_{1},\dots,d_{l}),\delta_{\otimes}^{BM})$ is
homeomorphic to the orbit space
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$
(see [21, Corollary 4.9]). For this reason, we also refer to this orbit space
as the compactum of tensorial bodies.

In [1, 2, 4] it is proved that $\mathcal{B}(d)$ is a $GL(d)$-proper space for
which $L(d)$ is a compact $O(d)$-global slice. This yields to new topological
models for $\mathcal{BM}(d),$ which turns out to be homeomorphic to the orbit
space $L(d)/O(d).$ Since $L(d)$ and $\mathcal{Q}$ are homeomorphic, a
representation of $\mathcal{BM}(d)$ as an orbit space $\mathcal{Q}/O(d)$ is
obtained. In the same spirit, in the context of tensorial bodies, we know that
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ is a $GL_{\otimes}$-proper space
for which $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ is a compact
$O_{\otimes}$-global slice, and the corresponding orbit spaces,
$\mathcal{L}_{\otimes}/O_{\otimes}$ and $\mathcal{B}_{\otimes}/GL_{\otimes},$
are homeomorphic [21]. Moreover, by [21, Corollary 4.9] and Theorem 5.4, the
compactum $\mathcal{BM}_{\otimes}(d_{1},\dots,d_{l})$ can be represented as an
orbit space $\mathcal{Q}/O_{\otimes}(d_{1},\dots,d_{l}).$

In the following proposition we show that, as occurs with the Banach-Mazur
compactum, the compactum of tensorial bodies is contractible.

######  Proposition 6.1.

Let $d_{i}\geq 2,$ $i=1,\dots,l,$ $\mathcal{BM}_{\otimes}(d_{1},\dots,d_{l})$
is contractible.

###### Proof.

We will define an $O_{\otimes}$-homotopy from
$\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ to the $O_{\otimes}$-fixed point
$B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}},$ as a consequence
the orbit space $\mathcal{L}_{\otimes}/O_{\otimes}$ will be contractible to
the point $\\{B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}}\\}.$
The result then will follow from the fact that $\mathcal{BM}_{\otimes}$ is
homeomorphic $\mathcal{L}_{\otimes}/O_{\otimes}$ ([21, Corollary 4.9]). Let
$H:\mathcal{L}_{\otimes}\times[0,1]\rightarrow\mathcal{L}_{\otimes}$ be the
$GL_{\otimes}$-homotopy used in the proof of Proposition 5.1-1, and let
$F:\Pi\cap\mathcal{L}_{\otimes}\times[0,1]\rightarrow\Pi\cap\mathcal{L}_{\otimes}$
be the $O_{\otimes}$-homotopy used in the proof of Proposition 5.1-3. Then,
the map $G:\mathcal{L}_{\otimes}\times[0,1]\rightarrow\mathcal{L}_{\otimes}$
defined as

$G(P,t)=\begin{cases}H(P,2t);\text{ if }t\in\left[0,\frac{1}{2}\right],\\\
F(conv_{\otimes}(P),2t-1);\text{ if
}t\in\left[\frac{1}{2},1\right].\end{cases}$

is the desired homotopy. Indeed, since
$H(P,1)=conv_{\otimes}(P)=F(conv_{\otimes}(P),0),$ $G$ is well-defined. Its
continuity and $O_{\otimes}$-equivariance follow directly from the properties
of $H$ and $F.$ Clearly, $G(P,0)=P$ and
$G(P,1)=B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}},$ for every
$Q\in\mathcal{L}_{\otimes}.$ ∎

In [3, Theorem 3.3], some conditions on the action of the orthogonal group
$O(d)$ on a Hilbert cube $X$ guaranteeing that the orbit space $X/O(d)$ is
homeomorphic to $\mathcal{BM}(d)$ are established. There an essential
hypothesis is that the action must have a unique $O(d)$-fixed point. In the
case of the compactum of tensorial bodies, represented as
$\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})/O_{\otimes}(d_{1},\dots,d_{l}),$ the
group $O_{\otimes}(d_{1},\dots,d_{l})$ is a maximal compact subgroup of
$GL_{\otimes}(d_{1},\dots,d_{l})$ consisting of orthogonal maps [21], for
which the $O_{\otimes}$-space $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l})$ has
at least three $O_{\otimes}$-fixed points.

######  Remark 6.2.

The Euclidean ball $B_{2}^{d_{1},\dots,d_{l}},$ and the tensor products
$B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}}$ and
$B_{2}^{d_{1}}\otimes_{\epsilon}\cdots\otimes_{\epsilon}B_{2}^{d_{l}}$ are
$O_{\otimes}$-fixed points in $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l}).$

###### Proof.

First notice that
$B_{2}^{d_{1},\dots,d_{l}},B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}}$
and $B_{2}^{d_{1}}\otimes_{\epsilon}\cdots\otimes_{\epsilon}B_{2}^{d_{l}}$ are
tensorial bodies w.r.t. $B_{2}^{d_{1}},\dots,B_{2}^{d_{l}},$ then, by [21,
Proposition 4.4 (2)],

$\ell_{\otimes}(B_{2}^{d_{1},\dots,d_{l}})=\ell_{\otimes}(B_{2}^{d_{1}}\otimes_{\epsilon}\cdots\otimes_{\epsilon}B_{2}^{d_{l}})=\ell_{\otimes}(B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}})=B_{2}^{d_{1},\dots,d_{l}}.$

Hence, all of them belong to $\mathcal{L}_{\otimes}(d_{1},\dots,d_{l}).$ Since
$U\in O_{\otimes}(d_{1},\dots,d_{l})$ is an orthogonal map
$U(B_{2}^{d_{1},\dots,d_{l}})=B_{2}^{d_{1},\dots,d_{l}},$ and thus
$B_{2}^{d_{1},\dots,d_{l}}$ is an $O_{\otimes}$-fixed point. On the other
hand, since any $U\in O_{\otimes}(d_{1},\dots,d_{l})$ can be written as
$U=(U_{1}\otimes\cdots\otimes U_{l})U_{\sigma},$ for some $U_{i}\in O(d_{i}),$
$i=1,\dots,l,$ and $U_{\sigma}\in\mathcal{P}$ [21, Proposition A.4], then

$\displaystyle
U(B_{2}^{d_{1}}\otimes_{\alpha}\cdots\otimes_{\alpha}B_{2}^{d_{l}})=U(B_{2}^{d_{1}}\otimes_{\alpha}\cdots\otimes_{\alpha}B_{2}^{d_{l}})$
$\displaystyle=U_{1}(B_{2}^{d_{\sigma(1)}})\otimes_{\alpha}\cdots\otimes_{\alpha}U_{l}(B_{2}^{d_{\sigma(l)}})$
$\displaystyle=B_{2}^{d_{\sigma(1)}}\otimes_{\alpha}\cdots\otimes_{\alpha}B_{2}^{d_{\sigma(l)}}$

for $\alpha=\pi,\epsilon,$ see equality (2.5). Now, given that the permutation
on the dimensions $d_{i}$’s is only possible if some of them are the same, we
must have that
$B_{2}^{d_{\sigma(1)}}\otimes_{\alpha}\cdots\otimes_{\alpha}B_{2}^{d_{\sigma(l)}}=B_{2}^{d_{1}}\otimes_{\alpha}\cdots\otimes_{\alpha}B_{2}^{d_{l}}$
and
$U(B_{2}^{d_{1}}\otimes_{\alpha}\cdots\otimes_{\alpha}B_{2}^{d_{l}})=B_{2}^{d_{1}}\otimes_{\alpha}\cdots\otimes_{\alpha}B_{2}^{d_{l}}$
for $\alpha=\pi,\epsilon$ and any $U\in O_{\otimes},$ as desired. ∎

Taking into account that the $GL_{\otimes}$-spaces
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ and $\Pi(d_{1},\dots,d_{l})$ are
homeomorphic $\mathcal{Q}$-manifolds (Corollary 5.5), the following natural
question arises:

Question 1 Are
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$ and
$\Pi(d_{1},\dots,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$ homeomorphic spaces?
In particular: Is the compactum $\mathcal{BM}_{\otimes}(d_{1},\dots,d_{l})$
homeomorphic to $\Pi(d_{1},\dots,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$?

Bringing together Proposition 3.1 and Corollary 5.5, we have that the
$GL_{\otimes}$-map
$conv_{\otimes}:\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})\rightarrow\Pi(d_{1},\dots,d_{l})$
is indeed a cell-like map between $\mathcal{Q}$-manifolds. In consequence, it
is also a near-homeomorphism [15, Corollary 43.2]. This motivates the
following question:

Question 2 Does there exist a $GL_{\otimes}$-homeomorphism between
$\mathcal{B}_{\otimes}(d_{1},\dots,d_{l})$ and $\Pi(d_{1},\dots,d_{l})$?

Clearly, a positive answer to the last question will provide a positive answer
to the first one. We finish the paper by exhibiting the relation between the
Banach-Mazur compactum and the orbit space
$\Pi(d_{1},\dots,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$ when the dimensions
$d_{i}$’s are different from each other.

######  Proposition 6.3.

The orbit space $\Pi(d_{1},\dots,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$ is
homeomorphic to the product
$\mathcal{BM}(d_{1})\times\cdots\times\mathcal{BM}(d_{l}),$ as long as
$d_{i}\neq d_{j}$ for $i\neq j,$ $i,j=1,\dots,l.$

###### Proof.

Consider the tensor product
$\otimes_{\pi}:\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l})\rightarrow\Pi(d_{1},\dots,d_{l}).$
We already know that $\otimes_{\pi}$ is a continuous surjective map [18,
Proposition 3.3]. Moreover, for every $(P_{1},\dots,P_{l})$ and
$(R_{1},\dots,R_{l})$ in
$\mathcal{B}(d_{1})\times\cdots\times\mathcal{B}(d_{l}),$ if $P_{i}$ belongs
to the $GL(d_{i})$-orbit of $R_{i},$ $i=1,\ldots,l,$ then $P_{i}=T_{i}R_{i}$
for some $T_{i}\in GL(d_{i}),$ $i=1,\dots,l,$ and

$P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}=T_{1}\otimes\cdots\otimes
T_{l}(R_{1}\otimes_{\pi}\cdots\otimes_{\pi}R_{l}).$

See equation (2.5). Hence, $P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}$
belongs to the $GL_{\otimes}$-orbit of
$R_{1}\otimes_{\pi}\cdots\otimes_{\pi}R_{l}.$ On the other side, since
$d_{i}\neq d_{j},$ for $i\neq j,$ then any map $T\in
GL_{\otimes}(d_{1},\dots,d_{l})$ can be written as
$T=T_{1}\otimes\cdots\otimes T_{l}$ for some $T_{i}\in GL(d_{i}),$
$i=1,\dots,l$ (see Section 1.2). This, together with equation (2.5), shows
that if $P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}$ belongs to the
$GL_{\otimes}$-orbit of $R_{1}\otimes_{\pi}\cdots\otimes_{\pi}R_{l},$ then
there exists $T_{i}\in GL(d_{i}),$ $i=1,\dots,l$ such that
$T_{1}R_{1}\otimes_{\pi}\cdots\otimes_{\pi}T_{l}R_{l}=P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}.$
Thus, from [18, Proposition 3.6], there exists $\lambda_{i}>0,$ $i=1,\dots,l,$
such that $\lambda_{1}\cdots\lambda_{l}=1$ and $T_{i}R_{i}=\lambda_{i}P_{i}$
for every $i.$ Thus, by the identity (3.1),

$P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}=\lambda_{1}P_{1}\otimes_{\pi}\cdots\otimes_{\pi}\lambda_{l}P_{l}=T_{1}\otimes\cdots\otimes
T_{l}(R_{1}\otimes_{\pi}\cdots\otimes_{\pi}R_{l}).$

Therefore, $P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}$ belongs to the
$GL_{\otimes}$-orbit of $R_{1}\otimes_{\pi}\cdots\otimes_{\pi}R_{l}$ if and
only if each $P_{i}$ belongs to the $GL(d_{i})$-orbit of $R_{i}.$ In
consequence, the map $\otimes_{\pi}$ induces a homeomorphism between the orbit
space $\Pi(d_{1},\dots,d_{l})/GL_{\otimes}(d_{1},\dots,d_{l})$ and
$\mathcal{BM}(d_{1})\times\cdots\times\mathcal{BM}(d_{l}).$ ∎

## References

  * [1] Antonyan, S. The topology of the Banach-Mazur compactum.  Fund. Math. 166, 3 (2000), 209–232.
  * [2] Antonyan, S. West’s problem on equivariant hyperspaces and Banach-Mazur compacta.  Trans. Amer. Math. Soc. 355, 8 (2003), 3379–3404.
  * [3] Antonyan, S. New topological models for Banach-Mazur compacta.  Fundam. Prikl. Mat. 11, 5 (2005), 19–31.
  * [4] Antonyan, S. Corrigendum to: “West’s problem on equivariant hyperspaces and Banach-Mazur compacta” [Trans. Amer. Math. Soc. 355 (2003), no. 8, 3379–3404; mr1974693].  Trans. Amer. Math. Soc. 358, 12 (2006), 5631–5633.
  * [5] Antonyan, S., and Jonard-Pérez, N. Affine group acting on hyperspaces of compact convex subsets of $\mathbb{R}^{n}$.  Fund. Math. 223 (2013), 99–136.
  * [6] Antonyan, S., Jonard-Pérez, N., and Juárez-Ordóñez, S. Hyperspaces of convex bodies of constant width.  347–361.
  * [7] Aubrun, G., Lami, L., Palazuelos, C., Szarek, S., and Winter, A. Universal gaps for XOR games from estimates on tensor norm ratios.  Comm. Math. Phys. 375, 1 (2020), 679–724.
  * [8] Aubrun, G., and Szarek, S. Tensor products of convex sets and the volume of separable states on n qudits.  Physical Review A 73, 2 (2006), 022109.
  * [9] Aubrun, G., and Szarek, S. Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory, vol. 223.  American Mathematical Soc., 2017.
  * [10] Bazilevich, L. Topology of a hyperspace of convex bodies of constant width.  Math. Notes 62, 6 (1997), 683–697.
  * [11] Bazylevych, L., and Zarichnyi, M. On convex bodies of constant width.  Topology Appl. 153, 11 (2006), 1699–1704.
  * [12] Belegradek, I. Hyperspaces of smooth convex bodies up to congruence.  Adv. Math. 332 (2018), 176–198.
  * [13] Bessaga, C. and Pełczyński, A. Selected Topics in Infinite-Dimensional Topology.  PWN—Polish Scientific Publishers, Warsaw, 1975.
  * [14] Bredon, G. Introduction to Compact Transformation Groups.  Academic Press, New York-London, 1972.
  * [15] Chapman, T. Lectures on Hilbert Cube Manifolds.  American Mathematical Society, Providence, R. I., 1976.  Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975, Regional Conference Series in Mathematics, No. 28.
  * [16] Defant, K., and Floret, K. Tensor Norms and Operator Ideals.  North Holland Mathematics Studies, 1992.
  * [17] Fernández-Unzueta, M. The Segre cone of Banach spaces and multilinear mappings.  Linear Multilinear Algebra (2018).  10.1080/03081087.2018.1509938.
  * [18] Fernández-Unzueta, M., and Higueras-Montaño, L. Convex Bodies Associated to Tensor Norms.  J. Convex Anal. 26, 4 (2019), 1297–1320.
  * [19] Fernández-Unzueta, M., and Higueras-Montaño, L. A general theory of tensor products of convex sets in Euclidean spaces.  Positivity 24, 5 (2020), 1373–1398.
  * [20] Gruber, P. The Space of Convex Bodies. In Handbook of Convex Geometry Vol. A, B.  North-Holland, Amsterdam, 1993, 301–318.
  * [21] Higueras-Montaño, L. A hyperspace of convex bodies arising from tensor norms.  Topology and Appl. 275 (2020), 107149, 20.
  * [22] Kadison, R., and Ringrose, J. Fundamentals of the Theory of Operator Algebras.  Academic Press, 1983.
  * [23] Macbeath, A. A compactness theorem for affine equivalence-classes of convex regions.  Canad. J. Math. 3 (1951), 54–61.
  * [24] van Mill, J. Infinite-Dimensional Topology: Prerequisites and Introduction. North-Holland Publishing Co., Amsterdam, 1989.
  * [25] Nadler, S., Quinn, J., and Stavrakas, N. Hyperspaces of compact convex sets.  Pacific J. Math. 83, 2 (1979), 441–462.
  * [26] Lindenstrauss, J. and Milman, V. D. The local theory of normed spaces and its applications to convexity.  In Handbook of Convex Geometry, Vol. A, B. North-Holland, Amsterdam, 1993, 1149–1220.
  * [27] Palais, R. The classification of $G$-spaces.  Mem. Amer. Math. Soc. No. 36, 1960.
  * [28] Palais, R. On the existence of slices for actions of non-compact Lie groups.  Ann. of Math. (2) 73 (1961), 295–323.
  * [29] Pears, A.R. Dimension Theory of General Spaces. Cambridge University Press, Cambridge, England-New York-Melbourne, 1975.
  * [30] Ryan, R. Introduction to Tensor Products of Banach Spaces.  Springer Monographs in Mathematics, 2002.
  * [31] Schäffer, J. Inner diameter, perimeter, and girth of spheres.  Math. Ann. 173 (1967), 59–79; addendum, ibid. 173 1967 79–82.
  * [32] Schneider, R. Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge, 1993.
  * [33] Thompson, A.C. Minkowski Geometry.  Cambridge University Press, Cambridge, 1996.
  * [34] Tomczak-Jaegermann, N. Banach-Mazur Distances and Finite-Dimensional Operator Ideals.  Longman Sc & Tech, 1989.
  * [35] Toruńczyk, H. On ${\rm CE}$-images of the Hilbert cube and characterization of $Q$-manifolds.  Fund. Math. 106, 1 (1980), 31–40.