# Convex Equipartitions inspired by the little cubes operad

Pavle V. M. Blagojević  Inst. Math., FU Berlin, Arnimallee 2, 14195 Berlin,
Germany  Mat. Institut SANU, Knez Mihailova 36, 11001 Beograd, Serbia
<EMAIL_ADDRESS>and  Nikola Sadovek  Inst. Math., FU Berlin,
Arnimallee 2, 14195 Berlin, Germany<EMAIL_ADDRESS>
###### Abstract.

A decade ago two groups of authors, Karasev, Hubard & Aronov and Blagojević &
Ziegler, have shown that the regular convex partitions of a Euclidean space
into $n$ parts yield a solution to the generalised Nandakumar & Ramana-Rao
conjecture when $n$ is a prime power. This was obtained by parametrising the
space of regular equipartitions of a given convex body with the classical
configuration space.

Now, we repeat the process of regular convex equipartitions many times, first
partitioning the Euclidean space into $n_{1}$ parts, then each part into
$n_{2}$ parts, and so on. In this way we obtain iterated convex equipartions
of a given convex body into $n=n_{1}\cdots n_{k}$ parts. Such iterated
partitions are parametrised by the (wreath) product of classical configuration
spaces. We develop a new configuration space – test map scheme for solving the
generalised Nandakumar & Ramana-Rao conjecture using the Hausdorff metric on
the space of iterated convex equipartions.

The new scheme yields a solution to the conjecture if and only if all the
$n_{i}$’s are powers of the same prime. In particular, for the failure of the
scheme outside prime power case we give three different proofs.

The research by Pavle V. M. Blagojević leading to these results has received
funding from the Serbian Ministry of Science, Technological development and
Innovations.

##  1\. Introduction and statement of the main result

The following intriguing question was asked by Nandakumar & Ramana-Rao in a
Nandakumar’s blog post [20] from 2006.

######  Conjecture 1.1 (Nandakumar & Ramana-Rao).

Every convex polygon $P$ in the plane can be partitioned into any prescribed
number $n$ of convex pieces that have equal area and equal perimeter.

Nandakumar & Ramana-Rao [21] gave an answer in case $n=2$ using the
intermediate value theorem and proposed a proof for the case $n=2^{k}\geq 4$.
Bárány, Blagojević & Szűcs [2, Thm. 1.1] gave the positive answer for the case
$n=3$ by setting an appropriate configuration space – test map (CS–TM) scheme
which allowed efficient used of Fadell–Husseini ideal valued index theory.
Soberón [25, Thm. 1] solved a similar question, namely the Bárány’s conjecture
of equally partitioning $d$ measures in $\mathbb{R}^{d}$ into $k$ pieces,
using Dold’s theorem [10]. Naturally, a higher dimensional extension of
Conjecture 1.1 was soon after formulated in [17, Thm. 1.3], [7, Thm. 1.3].

######  Conjecture 1.2 (Generalised Nandakumar & Ramana-Rao).

Let $K$ be a $d$-dimensional convex body in $\mathbb{R}^{d}$, let $\mu$ be an
absolutely continuous probability measure on $\mathbb{R}^{d}$, let $n\geq 2$
be any natural number, and let $\varphi_{1},\dots,\varphi_{d-1}$ be any $d-1$
continuous functions on the metric space of $d$-dimensional convex bodies in
$\mathbb{R}^{d}$. Then there exists a partition of $\mathbb{R}^{d}$ into $n$
convex pieces $P_{1},\dots,P_{n}$ such that equalities

$\mu(P_{1}\cap K)=\dots=\mu(P_{n}\cap K)$

and

$\varphi_{i}(P_{1}\cap K)=\dots=\varphi_{i}(P_{n}\cap K)$

hold for every $1\leq i\leq d-1$.

In the previous statement, absolute continuity of probability measure in
$\mathbb{R}^{d}$ is meant with respect to the Lebesgue measure on
$\mathbb{R}^{d}$, and the set of $d$-dimensional convex bodies in
$\mathbb{R}^{d}$ is endowed with the Hausdorff metric.

Karasev, Hubard & Aronov [17] proposed a solution for the generalised
Nandakumar & Ramana-Rao problem using a CS–TM scheme based on the generalised
Voronoi diagrams. The scheme was used to show that Conjecture 1.2 follows from
the non-existence of an $\mathfrak{S}_{n}$-equivariant map

$F(\mathbb{R}^{d},n)\longrightarrow S(W_{n}^{\oplus d-1}),$ (1)

where the symmetric group $\mathfrak{S}_{n}$ acts on $F(\mathbb{R}^{d},n)$ and
$W_{n}=\\{x\in\mathbb{R}^{n}:x_{1}+\dots+x_{n}=0\\}$ by permuting the points
and coordinates, respectively. Using a homological analogue of obstruction
theory, they showed [17, Thm. 1.10] non-existence of the map (1) when $n$ is a
prime power, and so gave the positive answer to the conjecture in this case.

Blagojević & Ziegler [7, Thm. 1.2] showed the non-existence of a map (1) if
and only if $n$ is a prime power using equivariant obstruction theory. In
order to apply the method, they developed a Salvetti-type CW-model
$\mathscr{F}(d,n)$ of the configuration space $F(\mathbb{R}^{d},n)$, which is
moreover its equivariant deformation retract. Resistance of Conjecture 1.2 to
the existing topological methods is manifested in the fact that the map (1)
exists whenever $n$ is not a prime power.

Blagojević, Lück & Ziegler [5, Thm. 8.3] showed the non-existence of a map (1)
for $n$ a prime using an analogue of Cohen’s vanishing theorem [9, Thm. 8.2].

In this paper we develop a cohomological method for identifying new classes of
solutions to the generalised Nandakumar & Ramana-Rao conjecture. The method
stems from a geometric idea of _iterated_ generalised Voronoi diagrams.
Namely, let $n=n_{1}\cdots n_{k}$ be a multiplicative decomposition of the
total number of parts in which we want to partition the convex body. In the
first iteration the convex body is partitioned into $n_{1}$ convex pieces of
equal area, and inductively, for each $2\leq i\leq k$, in the $i$th iteration
one further divides each of the $n_{1}\cdots n_{i-1}$ pieces into $n_{i}$
convex pieces of equal area. We call such partitions iterated of level $k$ and
type $(n_{1},\dots,n_{k})$. Our method is developed in three steps. We begin
by fixing a $d$-dimensional convex body $K$ in $\mathbb{R}^{d}$

In the first step, presented in Section 2, we define a space
$C_{k}({d};{n_{1},\dots,n_{k}})$, called the wreath product of configuration
spaces, which is used to parametrise iterated partitions of level $k$ and type
$(n_{1},\dots,n_{k})$ of $K$. Namely, we produce a map

$C_{k}({d};{n_{1},\dots,n_{k}})~{}\longrightarrow~{}\operatorname{\mathrm{EMP}}_{\mu}(K,n),$
(2)

where the codomain is the (metric) space of equal mass partitions of $K$ into
$n$ pieces. Moreover, our setting has an appropriate symmetry of the iterated
semi-direct product $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$ of the symmetric
groups, and the map (2) is equivariant with respect to it. Now, by setting an
appropriate CS–TM scheme and using the parametrisation map (2), we show that
the existence of an iterated solution of level $k$ and type
$(n_{1},\dots,n_{k})$ to the general Nandakumar & Ramana-Rao problem follows
from non-existence of an $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant
map of the form

$C_{k}({d};{n_{1},\dots,n_{k}})~{}\longrightarrow~{}S(W_{k}({d};{n_{1},\dots,n_{k}})).$

Here $W_{k}({d};{n_{1},\dots,n_{k}})$ denotes the relevant
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-representation, and
$S(W_{k}({d};{n_{1},\dots,n_{k}}))$ is the unit sphere in it.

In the second step, which is “outsourced” to Section 6, we prove the result on
continuity of partitions. Namely, the CS–TM scheme from [17] and [7] used the
fact that for a $d$-dimensional convex body $K\subseteq\mathbb{R}^{d}$ there
exists a continuous $\mathfrak{S}_{n}$-equivariant map

$F(\mathbb{R}^{d},n)~{}\longrightarrow~{}\operatorname{\mathrm{EMP}}(K,n),\qquad
x~{}\longmapsto~{}(K_{1}(x),\dots,K_{n}(x)),$

where $(K_{1}(x),\dots,K_{n}(x))$ denotes the convex partition of $K$ into $n$
parts of equal measure obtained from the generalised Voronoi diagram with
sites encoded by $x\in F(\mathbb{R}^{d},n)$. The existence of such a map
follows from the theory of optimal transport. We generalise this fact and
prove that the map

$\mathscr{K}^{d}\times
F(\mathbb{R}^{d},n)~{}\longrightarrow~{}(\mathscr{K}^{d})^{\times
n},\qquad(K,x)~{}\longmapsto~{}(K_{1}(x),\dots,K_{n}(x))\in\operatorname{\mathrm{EMP}}(K,n)$

is continuous, where $\mathscr{K}^{d}$ denotes the space of all
$d$-dimensional convex bodies in $\mathbb{R}^{d}$ endowed with the Hausdorff
metric. Continuity of this map, stated in Theorem 6.6, is a crucial result
which justifies the continuity of the parametrisation map (2), and therefore
verifies the validity of the CS–TM scheme from Section 2.3.

In the final and third step, contained in Section 3, we prove the main
topological result of the paper using equivariant obstruction theory.

######  Theorem 1.3.

Let $k\geq 1$, $d\geq 2$, and $n_{1},\dots,n_{k}\geq 2$ be integers. Then, an
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map of the form

$C_{k}({d};{n_{1},\dots,n_{k}})\longrightarrow
S(W_{k}({d};{n_{1},\dots,n_{k}}))$ (3)

does not exist if and only if $n_{1},\dots,n_{k}$ are powers of the same prime
number.

Therefore, this result gives a complete answer to the question of
effectiveness of the introduced CS–TM scheme in the following sense:

  * –

If all the numbers $n_{1},\dots,n_{k}$ are powers of the same prime number,
then an $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map of the form (3)
does not exist. Consequently, the CS–TM scheme guaranties existence of an
iterated solution of Conjecture 1.2.

  * –

If $n_{1},\dots,n_{k}$ are not all powers of the same prime number, then there
exists an $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map of the form
(3), hence the CS–TM scheme is not able to give any insights into the
existence of a solution to Conjecture 1.2.

In addition, using alternative arguments, we give two more proofs of the
existence of the equivariant map in Theorem 1.3. More precisely, in Section 4
we prove the existence of an equivariant map (3) using the structural map of
the little cubes operad, and in Section 5 by appropriately adapting the trick
of Özaydin in [22].

A direct consequence of Theorem 1.3 is a new positive solution of the
generalised Nandakumar & Ramana-Rao conjecture.

######  Corollary 1.4 (Iterated solution to generalised Nandakumar & Ramana-
Rao).

Let $K$ be a $d$-dimensional convex body in $\mathbb{R}^{d}$, let $\mu$ be an
absolutely continuous probability measure on $\mathbb{R}^{d}$, let
$n_{1},\dots,n_{k}\geq 2$ be integers, and let
$\varphi_{1},\dots,\varphi_{d-1}$ be any $d-1$ continuous functions on the
metric space of $d$-dimensional convex bodies in $\mathbb{R}^{d}$. If
$n_{1},\dots,n_{k}$ are all powers of the same prime number, then there exists
an iterated partition of $K$ of level $k$ and type $(n_{1},\dots,n_{k})$ into
$n:=n_{1}\cdots n_{k}$ convex pieces $K_{1},\dots,K_{n}$ such that equalities

$\mu(K_{1})=\dots=\mu(K_{n})$

and

$\varphi_{i}(K_{1})=\dots=\varphi_{i}(K_{n})$

hold for every $1\leq i\leq d-1$.

Moreover, the proof method, as noted by Karasev [16, Thm. 1.6], implies also
the following result,

######  Corollary 1.5.

Let $K$ be a $d$-dimensional convex body in $\mathbb{R}^{d}$, let $\mu$ be an
absolutely continuous probability measure on $\mathbb{R}^{d}$, let
$n_{1},\dots,n_{k}\geq 2$ be integers, and let
$\varphi_{1},\dots,\varphi_{d-1}$ be any $d-1$ additive, continuous functions
on the metric space of $d$-dimensional convex bodies in $\mathbb{R}^{d}$. Then
there exists an iterated partition of $K$ of level $k$ and type
$(n_{1},\dots,n_{k})$ into $n:=n_{1}\cdots n_{k}$ convex pieces
$K_{1},\dots,K_{n}$ such that equalities

$\mu(K_{1})=\dots=\mu(K_{n})$

and

$\varphi_{i}(K_{1})=\dots=\varphi_{i}(K_{n})$

hold for every $1\leq i\leq d-1$.

The idea of iterated partitions seems to have appeared first in context of the
Gromov’s waist of the sphere theorem [13], [19]. There, Gromov [13, Thm.
4.4.A] considered partitions of the sphere into $2^{i}$ pieces, which were
parametrised by the wreath products of spheres. In order to obtain a slightly
different waist of the sphere result, Palić [23, Thm. 5.2.5] considered
partitions of the sphere into $p^{i}$ pieces indexed by the $i$th wreath
product of the configuration spaces on $p$ points. Iterated partitions of
Euclidean spaces appeared in the work of Blagojević & Soberón [6, Sec. 2],
where they were parametrised by the $i$th join of the configuration space. For
the purposes of this paper we use a more general version of the wreath product
of configuration spaces, in the sense that the number of points of the
configuration space in different levels do not need to coincide.

##  2\. Spaces of Equal Mass Partitions

Let us denote by $\mathscr{K}^{d}$ the space of all full dimensional convex
bodies in the Euclidean space $\mathbb{R}^{d}$ endowed with the Hausdorff
metric and let $K\in\mathscr{K}^{d}$ be a convex body. As in the statement of
Conjecture 1.2, let $\mu$ be a probability measure on $\mathbb{R}^{d}$ which
is absolutely continuous with respect to the Lebesgue measure.

For an integer $n\geq 2$ and a convex body $K\in\mathscr{K}^{d}$, let us say
that $(K_{1},\dots,K_{n})\in(\mathscr{K}^{d})^{\times n}$ form a convex
partition of $K$ into $n$ pieces if

  * –

$K=K_{1}\cup\dots\cup K_{n}$, and

  * –

$\operatorname{\mathrm{int}}(K_{i})\cap\operatorname{\mathrm{int}}(K_{j})=\emptyset$
for each $1\leq i<j\leq n$.

Note that, implicitly, $\operatorname{\mathrm{int}}(K_{i})\neq\emptyset$, for
all $1\leq i\leq n$, because we require that $K_{i}\in\mathscr{K}^{d}$.

######  Definition 2.1.

Let $n\geq 2$ be an integer, let $K\in\mathscr{K}^{d}$ be a convex body, and
let $\mu$ be a probability measure on $\mathbb{R}^{d}$ which is absolutely
continuous with respect to the Lebesgue measure. We define
$\operatorname{\mathrm{EMP}}_{\mu}(K,n)$ to be the space of all convex
partitions of $K$ into $n$ pieces $(K_{1},\dots,K_{n})\in\mathscr{K}^{d}$ such
that

$\mu(K_{1})=\dots=\mu(K_{n}),$

endowed with the product Hausdorff metric induced from
$(\mathscr{K}^{d})^{\times n}$.

In the case of the plane, that is $d=2$, the perimeter function induces an
$\mathfrak{S}_{n}$-equivariant map

$\operatorname{\mathrm{EMP}}_{\mu}(K,n)\longrightarrow W_{n}$

which maps a partition $(K_{1},\dots,K_{n})$ to the vector obtained from
$(\mathrm{perim}(K_{1}),\dots,\mathrm{perim}(K_{n}))$ by subtracting the
average sum of all coordinates from each individual coordinate. Thus, an equal
area partition $(K_{1},\dots,K_{n})\in\operatorname{\mathrm{EAP}}(K,n)$ of $K$
gives a solution to the Nandakumar & Ramana-Rao conjecture if and only if it
is mapped to the origin by the above map.

Analogously, in the general case $d\geq 2$, there exists a map

$\operatorname{\mathrm{EMP}}_{\mu}(K,n)\longrightarrow W_{n}^{\oplus d-1}$ (4)

induced by the functions
$\varphi_{1},\dots,\varphi_{d-1}\colon\mathscr{K}^{d}\longrightarrow\mathbb{R}$
from the statement of Conjecture 1.2, which hits the origin in $W_{n}^{\oplus
d-1}$ if and only if there is a solution to the generalised Nandakumar &
Ramana-Rao conjecture. See [17, Sec. 2] and [7, Sec. 2] for more details.

So far, not enough is known about the topology of the space
$\operatorname{\mathrm{EMP}}_{\mu}(K,n)$ to answer directly the question
whether any map of the form (4) hits the origin. However, such a question has
been answered by restricting attention to a subspace of the, so called,
regular partitions of $K$, these are the ones which arise from piecewise-
linear convex functions.

Namely, using the the theory of optimal transport (see [17, Sec. 2] and [7,
Sec. 2] for more details), one can parametrise regular convex partitions
$(K_{1},\dots,K_{n})$ of $K$ into pieces of equal area by the classical
configuration space $F(\mathbb{R}^{2},n)$ of $n$ pairwise distinct points in
the plane. For an illustration see Figure 1. More precisely, there exists an
$\mathfrak{S}_{n}$-equivariant map

$F(\mathbb{R}^{d},n)\longrightarrow\operatorname{\mathrm{EMP}}_{\mu}(K,n).$
(5)

One can think of the parametrisation (5) as a prescribed way to assign to any
$n$ pairwise distinct points in the plane, called the sites, a partition of
$K$ into $n$ convex pieces of equal area. In particular, if the image of the
composition

$F(\mathbb{R}^{d},n)\longrightarrow\operatorname{\mathrm{EMP}}_{\mu}(K,n)\longrightarrow
W_{n}^{\oplus d-1}$ (6)

of maps (5) and (4) contains the origin, there exists a regular solution to
Conjecture 1.1.

Figure 1. A point $(x_{1},x_{2},x_{3},x_{4})\in F(\mathbb{R}^{2},4)$ induces a
regular partition
$(K_{1},K_{2},K_{3},K_{4})\in\operatorname{\mathrm{EMP}}(K,4)$ of the convex
body $K$.

In [17, Thm. 1.10] it was shown that whenever $n$ is a power of prime, every
$\mathfrak{S}_{n}$-equivariant map of the form

$F(\mathbb{R}^{d},n)\longrightarrow W_{n}^{\oplus d-1}$

hits the origin, thus showing the existence of a solution to the (generalised)
Nandakumar & Ramana-Rao conjecture in this case which is regular. Moreover, in
[7, Thm. 1.2] it was shown that outside of the prime power case, there always
exists an $\mathfrak{S}_{n}$-equivariant map
$F(\mathbb{R}^{d},n)\longrightarrow W_{n}^{\oplus d-1}\setminus\\{0\\}$. This
last result used equivariant obstruction theory, and it showed the limits of
the above configuration space – test map scheme proposed by the map (6).

Known topological methods are able to give solutions to the Nandakumar &
Ramana-Rao conjecture in the form of regular partitions. Since the space
$\operatorname{\mathrm{EMP}}_{\mu}(K,n)$ of convex equipartitions of $K$ is
larger than the set of regular partitions of $K$, a natural question arises:
Are there solutions to the Nandakumar & Ramana-Rao conjecture which are not
regular?

In this section we develop a method to parametrise iterated partitions of a
given full-dimensional convex body in $\mathbb{R}^{d}$ into $n$ parts. Namely,
we parametrise partitions with $k$ iteration by a wreath product of
configuration spaces $C_{k}(d;n_{1},..,n_{k})$, where $n=n_{1}\cdots n_{k}$
and $n_{1},\dots,n_{k}\geq 2$.

The rest of the section is organised as follows. In Section 2.1 we give an
example of the parametrisation in the case when $d=2$ and $k=2$. Then, in
Section 2.2, we describe partitions of level $k$ and type
$(n_{1},\dots,n_{k})$. Parametrisation of such partitions by the space
$C_{k}(d;n_{1},..,n_{k})$ is presented in Section 2.3, where the configuration
space – test map scheme is constructed.

###  2.1. Iterated partitions: first example

We establish a new configuration space – test map scheme with the intention of
identifying a wider class of solutions to Conjecture 1.1. To this end, we
parametrise a class of possibly non-regular equal area partitions of $K$ into
$n$ convex pieces. In this section, we give an example of such a
parametrisation for iterated partition of level two.

Let $a\geq 2$ and $b\geq 2$ be natural numbers such that $n=ab$. Consider a
parametrisation of equal are partitions of $K$ into convex pieces

$F(\mathbb{R}^{2},a)^{\times b}\times
F(\mathbb{R}^{2},b)\longrightarrow\operatorname{\mathrm{EAP}}(K,n)$ (7)

obtained as the composition of the following two maps:

  1. –

The first map

$F(\mathbb{R}^{2},a)^{\times b}\times F(\mathbb{R}^{2},b)\longrightarrow
F(\mathbb{R}^{2},a)^{\times b}\times\operatorname{\mathrm{EAP}}(K,b)$

is the the product of the identity on $F(\mathbb{R}^{d},b)^{\times a}$ and the
parametrisation (5).

  2. –

The second map

$F(\mathbb{R}^{2},a)^{\times
b}\times\operatorname{\mathrm{EAP}}(K,b)\longrightarrow\operatorname{\mathrm{EAP}}(K,n)$

on each slice $F(\mathbb{R}^{d},a)^{\times b}\times\\{(K_{1},\dots,K_{b})\\}$
of the domain equals to the product of maps

$F(\mathbb{R}^{2},a)\longrightarrow\operatorname{\mathrm{EAP}}(K_{i},a)$

for $1\leq i\leq b$, followed by the inclusion
$\operatorname{\mathrm{EAP}}(K_{1},a)\times\dots\times\operatorname{\mathrm{EAP}}(K_{b},a)\subseteq\operatorname{\mathrm{EAP}}(K,n)$.
In Section 6 we discuss in more details the continuity of this map.

The map (7) gives a parametrisation of equal area partition of $K$ into $n$
convex pieces which arise in two iterations. Namely, in the first iteration we
divide $K$ into $b$ convex pieces $(K_{1},\dots,K_{b})$ by a point in
$F(\mathbb{R}^{2},b)$ according to the rule (5), and in the second iteration
we again use the rule (5) to divide each of the pieces $K_{1},\dots,K_{b}$ by
$b$ points in $F(\mathbb{R}^{2},a)$. This is an example of an iterated
partition of level two and type (a,b). For an illustration see Figure 2.

Figure 2. Partition of convex body of: $(a)$ level $1$ and type $2$, $(b)$
level $2$ and type $(3,2)$.

In Section 2.1 we present a more general approach and define iterated
partition of level $k$ and type $(n_{1},\dots,n_{k})$, for any multiplicative
decomposition $n=n_{1}\cdots n_{k}$ with $n_{1},\dots,n_{k}\geq 2$. The idea
is that in the first iteration one divides $K$ into $n_{1}$ convex pieces of
equal area using a point in $F(\mathbb{R}^{2},n_{1})$ according to the rule
(5). Continuing inductively, for each $2\leq i\leq k$, in the $i$th iteration
one further divides each of the $n_{1}\cdots n_{i-1}$ pieces into $n_{i}$
convex pieces of equal area by as many points in $F(\mathbb{R}^{2},n_{i})$,
again according to the rule (5).

Let us consider the level two example once again and discuss symmetries of the
parameter space. In this context, the natural group acting on the domain
$F(\mathbb{R}^{2},a)^{\times b}\times F(\mathbb{R}^{2},b)$ of the map (7) is
the semi-direct product $(\mathfrak{S}_{a})^{\times
b}\rtimes\mathfrak{S}_{b}$, where $\mathfrak{S}_{b}$ acts on the product
$(\mathfrak{S}_{a})^{\times b}$ by permuting the factors. More precisely, the
action is given by

$(\tau_{1},\dots,\tau_{b};\sigma)\cdot(y_{1},\dots,y_{b};x)=(\tau_{1}\cdot
y_{\sigma^{-1}(1)},\dots,\tau_{a}\cdot y_{\sigma^{-1}(b)};\sigma\cdot x),$ (8)

for $(\tau_{1},\dots,\tau_{b};\sigma)\in(\mathfrak{S}_{a})^{\times
b}\rtimes\mathfrak{S}_{b}$ and $(y_{1},\dots,y_{b};x)\in
F(\mathbb{R}^{2},a)^{\times b}\times F(\mathbb{R}^{2},b)$, where the symmetric
group acts on the configuration space by permuting the coordinates. This
action induces an action on the codomain in such a way that the
parametrisation map (7) becomes $((\mathfrak{S}_{a})^{\times
b}\rtimes\mathfrak{S}_{b})$-equivariant. Indeed, an element
$(\tau_{1},\dots,\tau_{b};\sigma)\in(\mathfrak{S}_{a})^{\times
b}\rtimes\mathfrak{S}_{b}$ acts on a partition
$(K_{i,1},\dots,K_{i,a})_{i=1}^{b}\in\operatorname{\mathrm{EAP}}(K,n)$ by

$(\tau_{1},\dots,\tau_{b};\sigma)\cdot(K_{i,1},\dots,K_{i,a})_{i=1}^{b}=(K_{\sigma^{-1}(i),\tau^{-1}(1)},\dots,K_{\sigma^{-1}(i),\tau^{-1}(a)})_{i=1}^{b}.$

Thus, the expected full $\mathfrak{S}_{n}$-symmetry of the partition is
broken.

As above, the existence of a solution of the Nandakumar & Ramana-Rao
conjecture parametrised by (7) will be a consequence of the fact that the
composition of (7) with a perimeter-induced map into an
$((\mathfrak{S}_{a})^{\times b}\rtimes\mathfrak{S}_{b})$-representation hits
the origin. Let us define such a map

$\operatorname{\mathrm{EAP}}(K,n)\longrightarrow(W_{a})^{\oplus b}\oplus
W_{b}$ (9)

which sends a partition

$(K_{i,1},\dots,K_{i,a})_{i=1}^{b}\in\operatorname{\mathrm{EAP}}(K,n)$

to a vector which:

  * –

for each $1\leq i\leq b$, on the coordinates of the $i$th copy of $W_{a}$, has
the vector

$\Big{(}\mathrm{perim}(K_{i,1})-\frac{1}{a}\sum_{1\leq j\leq
a}\mathrm{perim}(K_{i,j}),\dots,\mathrm{perim}(K_{i,a})-\frac{1}{a}\sum_{1\leq
j\leq a}\mathrm{perim}(K_{i,j})\Big{)}\in W_{a},$

  * –

on the coordinates of $W_{b}$ has the vector

$\Big{(}\mathrm{perim}(K_{i,1})+\dots+\mathrm{perim}(K_{i,a})-\frac{1}{b}\sum_{1\leq
i\leq
b}\big{(}\mathrm{perim}(K_{i,1})+\dots+\mathrm{perim}(K_{i,a})\big{)}\Big{)}_{i=1}^{b}\in
W_{b}.$

In analogy to (8), the actions of $\mathfrak{S}_{b}$ and $\mathfrak{S}_{a}$ on
$W_{b}$ and $W_{a}$, respectively, induce an action of the semi-direct product
$(\mathfrak{S}_{a})^{\times b}\rtimes\mathfrak{S}_{b}$ on the vector space
$(W_{a})^{\oplus b}\oplus W_{b}$, making the map (9) equivariant with respect
to it.

Consequently, the $((\mathfrak{S}_{a})^{\times
b}\rtimes\mathfrak{S}_{b})$-equivariant composition

$F(\mathbb{R}^{2},a)^{\times b}\times
F(\mathbb{R}^{2},b)\longrightarrow\operatorname{\mathrm{EAP}}(P,n)\longrightarrow(W_{a})^{\oplus
b}\oplus W_{b}$

of (7) and (9) hits the origin, now in $(W_{a})^{\oplus b}\oplus W_{b}$, if
and only if there exists an iterated solution of level two and type $(a,b)$.
In Theorem 1.3 we prove that any $((\mathfrak{S}_{a})^{\times
b}\rtimes\mathfrak{S}_{b})$-equivariant map of the form

$F(\mathbb{R}^{2},a)^{\times b}\times
F(\mathbb{R}^{2},b)\longrightarrow(W_{a})^{\oplus b}\oplus W_{b}$

hits the origin if and only if $a$ and $b$ are powers of the same prime
number. Thus, the configuration space – test map scheme is able to give a
positive iterated solution of level two and type $(a,b)$ to the Nandakumar &
Ramana-Rao conjecture only in the case when $a$ and $b$ are powers of the same
prime.

###  2.2. Partition types

In this section, for integers $k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$, we
define iterated partitions of a given convex body $K\in\mathscr{K}^{d}$ which
are of level $k$ and type $(n_{1},\dots,n_{k})$. Furthermore we identify
natural groups of symmetries on such partition spaces.

For each convex body $K\in\mathscr{K}^{d}$ and each integer $n\geq 1$, there
exists an $\mathfrak{S}_{n}$-equivariant map

$F(\mathbb{R}^{d},n)\longrightarrow\operatorname{\mathrm{EMP}}(K,n)$ (10)

parametrising regular equal mass partitions of $K$ into $n$ convex pieces.
Here we denote by
$\operatorname{\mathrm{EMP}}(K,n)\subseteq(\mathscr{K}^{d})^{\times n}$ the
metric space of all equal mass partitions of $K$ into $n$ convex pieces. See
[17, Sec. 2] or [7, Sec. 2] for more details about this map.

As in Section 2.1, map (10) can be thought of as a prescribed way to divide
$K$ into $n$ convex parts of equal mass using $n$ pairwise distinct points in
$\mathbb{R}^{d}$. One can now repeat the process for each of the $n$ parts
individually, and divide them regularly into $m\geq 2$ parts, obtaining a
partition of $P$ into $nm$ parts, which is not necessarily regular. Continuing
this process, each of the $nm$ parts can be divided further, etc. In this
sense, regular partitions (10) can be thought of as partitions arising from a
single iteration, and in the following definition we introduce a notion of a
partition coming from arbitrarily many iterations. An illustration of the
iteration process see Figure 3.

######  Definition 2.2.

Let $K\in\mathscr{K}^{d}$ be a convex body.

  * –

(Iterated partition of level one) Let $n\geq 2$ be an integer. Any partition
of $K$ in the image of the map (10) is said to be _iterated of level $1$ and
type $n$_.

  * –

(Iterated partition of level $k\geq 2$) Suppose $k\geq 2$ and
$n_{1},\dots,n_{k}\geq 2$ are integers. Let

$(K_{1},\dots,K_{n_{1}\cdots
n_{k-1}})\in\operatorname{\mathrm{EMP}}(K,n_{1}\cdots n_{k-1}),$

be a partition of $K$ of level $k-1$ and type $(n_{1},\dots,n_{k-1})$, and let
$(K_{i,1},\dots,K_{i,n_{k}})$, for $1\leq i\leq n_{1}\cdots n_{k-1}$, be any
partition of $K_{i}$ of level $1$ and type $n_{k}$. Then the partition

$(K_{i,j}:1\leq i\leq n_{1}\cdots n_{k-1},~{}1\leq j\leq n_{k})$

of $K$ into $n_{1}\cdots n_{k}$ convex parts of equal mass is said to be
_iterated of level $k$ and type $(n_{1},\dots,n_{k})$_.

An iterated partition of level $k$ and type $(n_{1},\dots,n_{k})$ arises in
$k$ inductive steps. For every iteration $0\leq i\leq k-1$, each of the
$n_{k}\cdots n_{k-i+1}$ elements of the partition in the $i$th step contains
exactly $n_{k-i}$ elements in the partition from the $(i+1)$st step.
Therefore, the collection of all such intermediate parts forms a ranked poset
with respect to inclusion. More precisely:

  * –

The maximum of the poset is the full convex body $K$.

  * –

For each $1\leq i\leq k$ the elements of the poset which are of level $i$ are
precisely the elements of the partition which arise in the $i$th inductive
step.

For fixed parameters $k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$, the induced
posets arising from different convex bodies are isomorphic. Since the posets
are supposed to model the type of iterated division process, from now on we
will fix one poset of each type.

Figure 3. Partition of convex body of: $(a)$ level $1$ and type $2$, $(b)$
level $2$ and type $(3,2)$, level three and type $(2,3,2)$.

######  Definition 2.3.

Let $k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. Let us denote by
$P_{k}(n_{1},\dots,n_{k})$ a fixed poset, called poset of level $k$ and type
$(n_{1},\dots,n_{k})$, built inductively as follows.

  * –

(Case $k=1$) Let $P_{1}(n_{1})$ be a poset on $n_{1}+1$ elements
$\pi^{0},\pi_{1}^{1},\dots,\pi_{n_{1}}^{1}$ with relations

$\pi_{1}^{1},\dots,\pi_{n_{k}}^{1}\preccurlyeq\pi^{0}.$

The maximum $\pi^{0}$ is at the level zero and elements
$\pi_{1}^{1},\dots,\pi_{n_{1}}^{1}$ are at the level one.

  * –

(Case $k\geq 2$) Let the poset $P_{k}(n_{1},\dots,n_{k})$ be obtained from the
poset $P_{k-1}(n_{1},\dots,n_{k-1})$ by adding $n_{1}\cdots n_{k}$ elements
$\pi_{1}^{k},\dots,\pi_{n_{1}\cdots n_{k}}^{k}$ of level $k$ and relations

$\pi_{(i-1)n_{k}+1}^{k},\dots,\pi_{(i-1)n_{k}+n_{k}}^{k}\preccurlyeq\pi^{k-1}_{i}$

for each $1\leq i\leq n_{1}\cdots n_{k-1}$.

Notice that the Hasse diagram of the poset $P_{k}(n_{1},...,n_{k})$ is a
rooted tree with $k$ levels, where a vertex of level $0\leq i\leq k-1$ has
precisely $n_{k-i}$ “children”. For an example of posets
$P_{k}(n_{1},...,n_{k})$ see Figure 4.

Figure 4. Hasse diagram of posets: $(a)$ $P_{1}(2)$, $(b)$ $P_{2}(3,2)$, $(c)$
$P_{3}(2,3,2)$.

For integers $k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$, let us denote by
$\operatorname{\mathrm{Aut}}_{k}(n_{1},\dots,n_{k})$ the group of
automorphisms of the poset $P_{k}(n_{1},\dots,n_{k})$.

In the case $k=1$, it is useful to consider iterated partitions of level $1$
and type $n$ together with the group of symmetries of the poset $P_{1}(n)$. In
fact, the group of symmetries in this case is
$\operatorname{\mathrm{Aut}}_{1}(n)\cong\mathfrak{S}_{n}$ which coincides with
the natural group of symmetries of the configuration space
$F(\mathbb{R}^{d},n)$. This group was essentially used for identification of
partitions of level $1$ and type $n$ which solve the general Nandakumar &
Ramana-Rao conjecture [17], [7].

More generally, the group $\operatorname{\mathrm{Aut}}_{k}(n_{1},\dots,n_{k})$
naturally acts on the set of partitions of level $k$ and type
$(n_{1},\dots,n_{k})$, and will be crucial ingredient for the configuration
map – test map scheme set up in Section 2.3 to identify iterated solutions to
the general Nandakumar & Ramana-Rao conjecture which are of level $k$ and type
$(n_{1},\dots,n_{k})$.

######  Definition 2.4.

Let $k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. We define the group
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$ inductively as follows.

  * –

For $k=1$ we set $\mathscr{S}_{1}({n_{1}}):=\mathfrak{S}_{n_{1}}$.

  * –

For $k\geq 2$ let

$\mathscr{S}_{k}({n_{1},\dots,n_{k}}):=\mathscr{S}_{k-1}({n_{1},\dots,n_{k-1}})^{\times
n_{k}}\rtimes\mathfrak{S}_{n_{k}}$

be the semi-direct product; for a definition of semi-direct product consult
for example [11, Sec. 5.5]. The symmetric group $\mathfrak{S}_{n_{k}}$ acts on
the product $\mathscr{S}_{k-1}({n_{1},\dots,n_{k-1}})^{\times n_{k}}$ by

$\sigma\cdot(\Sigma_{1},\dots,\Sigma_{n_{k}})=(\Sigma_{\sigma^{-1}(1)},\dots,\Sigma_{\sigma^{-1}(n_{k})}),$

for $\sigma\in\mathfrak{S}_{n}$ and
$(\Sigma_{1},\dots,\Sigma_{n_{k}})\in\mathscr{S}_{k-1}({n_{1},\dots,n_{k-1}})^{\times
n_{k}}$.

Written down in more details for $k\geq 2$, the operation in the wreath
product $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$ is given inductively by

$(\Sigma_{1},\dots,\Sigma_{n_{k}};\sigma)\cdot(\Theta_{1},\dots,\Theta_{n_{k}};\theta)=(\Sigma_{1}\cdot\Theta_{\sigma^{-1}(1)},\dots,\Sigma_{n_{k}}\cdot\Theta_{\sigma^{-1}(n_{k})};\sigma\theta),$

where
$(\Sigma_{1},\dots,\Sigma_{n_{k}};\sigma),(\Theta_{1},\dots,\Theta_{n_{k}};\theta)\in\mathscr{S}_{k-1}({n_{1},\dots,n_{k-1}})^{\times
n_{k}}\rtimes\mathfrak{S}_{n_{k}}=\mathscr{S}_{k}({n_{1},\dots,n_{k}})$.

######  Lemma 2.5.

Let $k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. Then, there is a
group isomorphism

$\operatorname{\mathrm{Aut}}_{k}(n_{1},\dots,n_{k})~{}\cong~{}\mathscr{S}_{k}({n_{1},\dots,n_{k}}).$

###### Proof.

The proof is by induction on $k\geq 1$, becasue the definitions of relevant
objects were given inductively.

For $k=1$ we have that
$\operatorname{\mathrm{Aut}}_{1}(n_{1})\cong\mathfrak{S}_{n_{1}}$. Indeed,
$\sigma\in\mathfrak{S}_{n_{1}}$ acts on the level one elements of
$P_{1}(n_{1})$ by the rule $\pi_{i}^{1}\longmapsto\pi_{\sigma(i)}^{1}$ for
each $1\leq i\leq n$.

Suppose now that $k\geq 2$. By the induction hypothesis, it is enough to show
the existence of an isomorphism

$\operatorname{\mathrm{Aut}}_{k-1}(n_{1},\dots,n_{k-1})^{\times
n_{k}}\rtimes\mathfrak{S}_{n}\xrightarrow{~{}\cong~{}}\operatorname{\mathrm{Aut}}_{k}(n_{1},\dots,n_{k}).$
(11)

One way to construct an isomorphism is to map an element

$(\phi_{1},\dots,\phi_{n};\sigma)\in\operatorname{\mathrm{Aut}}_{k-1}(n_{1},\dots,n_{k-1})^{\times
n_{k}}\rtimes\mathfrak{S}_{n}$

by a composition of the following two maps.

  1. (1)

The first one permutes the level one elements $\pi^{1}_{i}$, together with
their respective principal ideals
$\langle\pi^{1}_{i}\rangle:=P_{k}(n_{1},\dots,n_{k})_{\preccurlyeq\pi_{i}^{1}}$,
by the rules

$\pi^{1}_{i}\longmapsto\pi^{1}_{\sigma(i)}~{}~{}\text{ and
}~{}~{}\langle\pi^{1}_{i}\rangle\xrightarrow{~{}\cong~{}}\langle\pi_{\sigma(i)}^{1}\rangle,$

for each $1\leq i\leq n_{k}$, where the (order) ideals are mapped in the
canonical way preserving the order of the indices on each level.

  2. (2)

The second one applies the automorphism $\phi_{i}$ to the (order) ideal
$\langle\pi_{i}^{1}\rangle$ using the unique isomorphism
$\langle\pi_{i}^{1}\rangle\cong P_{k-1}(n_{1},\dots,n_{k-1})$ which preserves
the order of the indices, for each $1\leq i\leq n_{k}$.

Defined in this way, the map (11) is a bijection, since each element of
$\operatorname{\mathrm{Aut}}_{k}(n_{1},\dots,n_{k})$ is uniquely determined by
the permutation of its level-one elements and the automorphism of their
respective principle (order) ideals. For an illustration of the isomorphism
(11) see Figure 5. ∎

Figure 5. Action of the element $((123),(13);(12))\in\mathfrak{S}_{3}$.

######  Definition 2.6.

Assume that a group $G$ acts on the set $X$ and that the symmetric group
$\mathfrak{S}_{n}$ acts on the set $Y$. Let $G^{\times
n}\rtimes\mathfrak{S}_{n}$ be the semi-direct product, where
$\mathfrak{S}_{n}$ acts on the product $G^{\times n}$ by permuting the
coordinates. The induced action of $G^{\times n}\rtimes\mathfrak{S}_{n}$ on
the set $X^{\times n}\times Y$, given by

$(g_{1},\dots,g_{n};\sigma)\cdot(x_{1},\dots,x_{n};y):=(g_{1}\cdot
x_{\sigma^{-1}(1)},\dots,g_{n}\cdot x_{\sigma^{-1}(n)};\sigma\cdot y)$

for each $(g_{1},\dots,g_{n})\in G^{\times n}$, $\sigma\in\mathfrak{S}_{n}$,
$(x_{1},\dots,x_{n})\in X^{\times n}$, and $y\in Y$, is called the wreath
product action.

###  2.3. Configuration Space – Test Map scheme

In the previous section we described partitions of $K$ into $n:=n_{1}\cdots
n_{k}$ parts of level $k$ and type $(n_{1},\dots,n_{k})$. Hence, all such
partitions are contained in the space $\operatorname{\mathrm{EMP}}(K,n)$, but
is it possible to continuously parametrise them by a topological space?

In this section we first give a positive answer to the previous question, and
then develop the related configuration space – test map scheme for finding
solutions to Conjecture 1.2 which are of level $k$ and type
$(n_{1},\dots,n_{k})$.

######  Definition 2.7 (Wreath product of Configuration Spaces).

Let $d,k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. The space
$C_{k}({d};{n_{1},\dots,n_{k}})$ is defined inductively as follows.

  * –

For $k=1$ set $C_{1}({d};{n_{1}}):=F(\mathbb{R}^{d},n_{1})$.

  * –

For $k\geq 2$ let

$C_{k}({d};{n_{1},\dots,n_{k}}):=C_{k-1}({d};{n_{1},\dots,n_{k-1}})^{\times
n_{k}}\times F(\mathbb{R}^{d},n_{k}).$

The action of the symmetric group $\mathfrak{S}_{n}$ on the configuration
space $F(\mathbb{R}^{d},n)$ induces an action of the group
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$ on $C_{k}({d};{n_{1},\dots,n_{k}})$.
Let us introduce this action explicitly by induction on $k\geq 1$.

  * –

For $k=1$ the group $\mathscr{S}_{1}({n})=\mathfrak{S}_{n}$ acts on
$C_{1}({d};{n})=F(\mathbb{R}^{d},n)$ by permuting the coordinates

$\sigma\cdot(x_{1},\dots,x_{n})=(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(n)}),$

where $\sigma\in\mathfrak{S}_{n}$ and $(x_{1},\dots,x_{n})\in
F(\mathbb{R}^{d},n)$.

  * –

For $k\geq 2$ and $n_{1},\dots,n_{k}\geq 2$ the group
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$ acts on the space
$C_{k}({d};{n_{1},\dots,n_{k}})$ by the rule

$(\Sigma_{1},\dots,\Sigma_{n_{k}};\sigma)\cdot(X_{1},\dots,X_{n_{k}};x)=(\Sigma_{1}\cdot
X_{\sigma^{-1}(1)},\dots,\Sigma_{n_{k}}\cdot
X_{\sigma^{-1}(n_{k})};\sigma\cdot x),$

where

$(\Sigma_{1},\dots,\Sigma_{n_{k}};\sigma)\in\mathscr{S}_{k-1}({n_{1},\dots,n_{k-1}})^{\times
n_{k}}\rtimes\mathfrak{S}_{n_{k}}=\mathscr{S}_{k}({n_{1},\dots,n_{k}})$

and

$(X_{1},\dots,X_{n_{k}};x)\in C_{k-1}({d};{n_{1},\dots,n_{k-1}})^{\times
n_{k}}\times F(\mathbb{R}^{d},n_{k})=C_{k}({d};{n_{1},\dots,n_{k}}).$

In similar situations, where the action is defined analogously, we will call
it simply the wreath product action, in the light of Definition 2.6.

Next, for a given convex body $K\in\mathscr{K}^{d}$, we define a
parametrisation of the iterated equal mass convex partitions of $K$.

######  Proposition 2.8 (Parametrisation of iterated partitions).

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers, and let
$K\in\mathscr{K}^{d}$ be a $d$-dimensional convex body. Then, there exists a
continuous $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$C_{k}({d};{n_{1},\dots,n_{k}})\longrightarrow\operatorname{\mathrm{EMP}}_{\mu}(K,n_{1}\cdots
n_{k}).$

###### Proof.

As already discussed in the case when $\mu$ is the Lebesgue measure itself,
for each integer $n\geq 2$ and $K\in\mathscr{K}^{d}$, due to the theory of
optimal transport (see [17, Sec. 2] and [7, Sec. 2]) there exists an
$\mathfrak{S}_{n}$-equivariant map

$F(\mathbb{R}^{d},n)\times\\{K\\}\longrightarrow\operatorname{\mathrm{EMP}}_{\mu}(K,n),~{}~{}~{}(x,K)\longmapsto(K_{1}(x),\dots,K_{n}(x)),$
(12)

where $(K_{1}(x),\dots,K_{n}(x))$ represents the unique regular equipartition
of $K$ with respect to the measure $\mu$ and sites $x=(x_{1},\dots,x_{n})\in
F(\mathbb{R}^{d},n)$. In Lemma 6.3 we provide a topological proof of the
existence of the map (12).

One can consider (12) as the map given on slices
$F(\mathbb{R}^{d},n)\times\\{K\\}\subseteq
F(\mathbb{R}^{d},n)\times\mathscr{K}^{d}$ and construct an
$\mathfrak{S}_{n}$-equivariant map with enlarged domain

$p:F(\mathbb{R}^{d},n)\times\mathscr{K}^{d}\longrightarrow(\mathscr{K}^{d})^{\times
n},~{}~{}~{}~{}~{}(x,K)\longmapsto(K_{1}(x),\dots,K_{n}(x))\in\operatorname{\mathrm{EMP}}_{\mu}(K,n),$
(13)

where the symmetric group acts on the $\mathscr{K}^{d}$-coordinate of the
domain trivially, and on the codomain by permuting the coordinates. Continuity
of the map (13) is proved in Theorem 6.6, and it, in particular, implies
continuity of the map (12).

We prove something a bit stronger than the statement of the proposition.
Namely, we show existence of an
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$p_{k}:C_{k}({d};{n_{1},\dots,n_{k}})\times\mathscr{K}^{d}\longrightarrow(\mathscr{K}^{d})^{\times
n_{1}\cdots n_{k}}$ (14)

which on each slice

$C_{k}({d};{n_{1},\dots,n_{k}})\times\\{K\\}\subseteq
C_{k}({d};{n_{1},\dots,n_{k}})\times\mathscr{K}^{d}$

equals to the desired $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$C_{k}({d};{n_{1},\dots,n_{k}})\times\\{K\\}\longrightarrow\operatorname{\mathrm{EMP}}_{\mu}(K,n_{1}\cdots
n_{k})\subseteq(\mathscr{K}^{d})^{\times n_{1}\cdots n_{k}}$

from the statement of this proposition. The
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-action on the
$\mathscr{K}^{d}$-coordinate of the domain of the map (14) is trivial, and the
action on the codomain of (14) is given inductively by the wreath product
action (see Definition 2.6).

The construction of a map $p_{k}$ is done by induction on $k\geq 1$. For the
base case $k=1$, the map (13) has the desired restriction properties (12),
hence we set $p_{1}:=p$. Let now $k\geq 2$ and set ${\bf
n}:=(n_{1},\dots,n_{k})$ and ${\bf n}^{\prime}:=(n_{1},\dots,n_{k-1})$ to
simplify the notation. Assume moreover there exists an
$\mathscr{S}_{k-1}({{\bf n}^{\prime}})$-equivariant parametrisation map

$p_{k-1}:C_{k-1}({d};{{\bf
n}^{\prime}})\times\mathscr{K}^{d}\longrightarrow(\mathscr{K}^{d})^{\times
n_{1}\cdots n_{k-1}}$ (15)

with the required slice-wise restrictions. The map $p_{k}$ is constructed from
two ingredients.

  1. (1)

Let the $\mathscr{S}_{k}({\bf n})$-equivariant map

$\operatorname{\mathrm{id}}\times p_{1}:C_{k-1}({d};{{\bf
n}^{\prime}})^{\times n_{k}}\times
F(\mathbb{R}^{d},n_{k})\times\mathscr{K}^{d}\longrightarrow C_{k-1}({d};{{\bf
n}^{\prime}})^{\times n_{k}}\times(\mathscr{K}^{d})^{\times n_{k}}$ (16)

be the product of the identity on $C_{k-1}({d};{{\bf n}^{\prime}})^{\times
n_{k}}$ and the map (13) with value $n=n_{k}$. The action of the group
$\mathscr{S}_{k}({{\bf n}})=\mathscr{S}_{k-1}({{\bf n}^{\prime}})^{\times
n_{k}}\rtimes\mathfrak{S}_{n_{k}}$ on the codomain of the map (16) is induced
by the product action of $\mathscr{S}_{k-1}({{\bf n}^{\prime}})^{\times
n_{k}}$ on $C_{k-1}({d};{{\bf n}^{\prime}})^{\times n_{k}}$, and the action of
$\mathfrak{S}_{n_{k}}$ on the product $(\mathscr{K}^{d})^{\times n_{k}}$ which
permutes the coordinates. Notice that the map (16) restricts to the slice-wise
equivariant map

$C_{k}({d};{{\bf n}})\times\\{K\\}\longrightarrow C_{k-1}({d};{{\bf
n}^{\prime}})^{\times
n_{k}}\times\operatorname{\mathrm{EMP}}_{\mu}(K,n_{k})\subseteq
C_{k-1}({d};{{\bf n}^{\prime}})^{\times n_{k}}\times(\mathscr{K}^{d})^{\times
n_{k}}$

for each $K\in\mathscr{K}^{d}$.

  2. (2)

The $n_{k}$-fold product $(p_{k-1})^{\times n_{k}}$ of the map (15) induces
the $\mathscr{S}_{k}({{\bf n},n_{k}})$-equivariant map

$C_{k-1}({d};{{\bf n}^{\prime}})^{\times n_{k}}\times(\mathscr{K}^{d})^{\times
n_{k}}\xrightarrow{~{}(p_{k-1})^{\times n_{k}}~{}}((\mathscr{K}^{d})^{\times
n_{1}\cdots n_{k-1}})^{\times n_{k}}.$ (17)

From the induction hypothesis it follows that the map (17) restricts to the
slise-wise equivariant map

$C_{k-1}({d};{{\bf n}^{\prime}})^{\times
n_{k}}\times\\{(K_{1},\dots,K_{n_{k}})\\}\longrightarrow\prod_{i=1}^{n_{k}}\operatorname{\mathrm{EMP}}_{\mu}(K_{i},n_{1}\cdots
n_{k-1})\subseteq\prod_{i=1}^{n_{k}}(\mathscr{K}^{d})^{\times n_{1}\cdots
n_{k-1}}$

for each $(K_{1},\dots,K_{n_{k}})\in(\mathscr{K}^{d})^{\times n_{k}}$.

Having these two ingredients, we can define the map (14) as the
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant composition

$p_{k}:C_{k}({d};{\bf n})\times\mathscr{K}^{d}\xrightarrow{~{}\eqref{eq: first
map in the partition map}~{}}C_{k-1}({d};{{\bf n}^{\prime}})^{\times
n_{k}}\times(\mathscr{K}^{d})^{\times n_{k}}\xrightarrow{~{}\eqref{eq: second
map in the partition map}~{}}(\mathscr{K}^{d})^{\times n_{1}\cdots n_{k}}.$

The slice-wise restrictions of the maps (16) and (17) imply that the
restriction of the map $p_{k}$ factors as

$C_{k}({d};{\bf
n})\times\\{K\\}\longrightarrow\operatorname{\mathrm{EMP}}_{\mu}(K,n_{1}\cdots
n_{k})$

for each $K\in\mathscr{K}^{d}$, which finishes the proof. ∎

In order to proceed with setting up the configuration space – test map scheme,
we first define an $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-representation which
is used as a codomain of the test map of the scheme.

For each $n\geq 1$ let
$W_{n}:=\\{x\in\mathbb{R}^{n}:~{}x_{1}+\dots+x_{n}=0\\}$. It will be
convenient to consider $W_{n}$ to be a space of row vectors with zero
coordinate sums.

######  Definition 2.9.

Let $d\geq 1$, $k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. The vector
space $W_{k}({d};{n_{1},\dots,n_{k}})$ is defined inductively as follows.

  * –

For $k=1$ set

$W_{1}({d};{n_{1}}):=W_{n_{1}}^{\oplus
d-1}\cong\\{(z_{1},\dots,z_{n_{1}})\in(\mathbb{R}^{d-1})^{\oplus
n_{1}}:~{}z_{1}+\dots+z_{n_{1}}=0\\}.$

  * –

For $k\geq 2$ let

$W_{k}({d};{n_{1},\dots,n_{k}}):=W_{k-1}({d};{n_{1},\dots,n_{k-1}})^{\oplus
n_{k}}\oplus W_{n_{k}}^{\oplus d-1}.$

The dimension of the vector space in the case $k=1$ is $\dim
W_{1}({d};{n_{1}})=(d-1)(n_{1}-1)$. From the inductive relation

$\dim\big{(}W_{k}({d};{n_{1},\dots,n_{k}})\big{)}=n_{k}\cdot\dim\big{(}W_{k-1}({d};{n_{1},\dots,n_{k-1}})\big{)}+(d-1)(n_{k}-1),$

where $k\geq 2$, it follows that

$\dim\big{(}W_{k}({d};{n_{1},\dots,n_{k}})\big{)}=(d-1)(n_{1}\cdots n_{k}-1).$

In the analogy to the case of $C_{k}({d};{n_{1},\dots,n_{k}})$, the action of
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$ on $W_{k}({d};{n_{1},\dots,n_{k}})$ is
defined by the wreath product action; consult Definition 2.6.

Let $S(W_{k}({d};{n_{1},\dots,n_{k}}))$ be the unit sphere of
$W_{k}({d};{n_{1},\dots,n_{k}})$. Now, we state the main theorem of this
section which establishes the configuration space – test map scheme.

######  Theorem 2.10.

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. Let
$K\in\mathscr{K}^{d}$ be a $d$-dimensional convex body and $n:=n_{1}\cdots
n_{k}$. If there is no $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map
of the form

$C_{k}({d};{n_{1},\dots,n_{k}})\longrightarrow
S(W_{k}({d};{n_{1},\dots,n_{k}})),$ (18)

then there exists a solution to Conjecture 1.2 for the convex body $K$ which
is of level $k$ and type $(n_{1},\dots,n_{k})$.

###### Proof.

Let us first construct an $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant
map

$\Phi_{k}:\operatorname{\mathrm{EMP}}_{\mu}(K,n_{1}\cdots
n_{k})\longrightarrow W_{k}({d};{n_{1},\dots,n_{k}}),$ (19)

which tests whether a convex equipartition of $K$ into $n_{1}\cdots n_{k}$
parts is a solution to Conjecture 1.2. The map $\Phi_{k}$ will be induced from
the functions
$\varphi_{1},\dots,\varphi_{d-1}:\mathscr{K}^{d}\rightarrow\mathbb{R}$, which
are given in the statement of the conjecture, and will be defined inductively
on $k\geq 1$. In fact, it will be given on a larger domain
$(\mathscr{K}^{d})^{\times n_{1}\cdots n_{k}}$, where the action of the group
$\mathscr{S}_{k}({n_{1},...,n_{k}})$ is as before.

For $k=1$, the $\mathfrak{S}_{n_{1}}$-equivariant map

$\Phi_{1}:(\mathscr{K}^{d})^{\times n_{1}}\longrightarrow W_{n_{1}}^{\oplus
d-1}$

is given by mapping $(K_{1},...,K_{n_{1}})$ to the an element which, for each
$1\leq i\leq d-1$, has in the coordinate living in the $i$th copy of
$W_{n_{1}}$ the vector obtained from

$\Big{(}\varphi_{i}(K_{1})-\frac{1}{n_{1}}\big{(}\varphi_{i}(K_{1})+\dots+\varphi(K_{n_{1}})\big{)},\dots,\varphi_{i}(K_{n_{1}})-\frac{1}{n_{1}}\big{(}\varphi_{i}(K_{1})+\dots+\varphi(K_{n_{1}})\big{)}\Big{)}\in
W_{n_{1}}.$

Suppose $k\geq 2$ and assume the
$\mathscr{S}_{k-1}({n_{1},\dots,n_{k-1}})$-equivariant map

$\Phi_{k-1}:(\mathscr{K}^{d})^{\times n_{1}\cdots n_{k-1}}\longrightarrow
W_{k-1}({d};{n_{1},\dots,n_{k-1}})$

is already constructed . We define the desired
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$\Phi_{k}:((\mathscr{K}^{d})^{\times n_{1}\cdots n_{k-1}})^{\times
n_{k}}\longrightarrow W_{k-1}({d};{n_{1},\dots,n_{k-1}})^{\oplus n_{k}}\oplus
W_{n_{k}}^{\oplus d-1}$

by mapping an element $(\Pi_{1},...,\Pi_{n_{k}})$ to an element which:

  * –

for each $1\leq j\leq n_{k}$, at the coordinate living in the $j$th copy of
$W_{k-1}({d};{n_{1},\dots,n_{k-1}})$ has the value $\Phi_{k-1}(\Pi_{j})$, and

  * –

for each $1\leq i\leq d-1$, at the coordinate living in the $i$th copy of
$W_{n_{k}}$ is the vector

$\Big{(}\varphi^{\Sigma}_{i}(\Pi_{1})-\frac{1}{n_{k}}\big{(}\varphi^{\Sigma}_{i}(\Pi_{1})+\dots+\varphi^{\Sigma}_{i}(\Pi_{n_{k}})\big{)},\dots,\varphi^{\Sigma}_{i}(\Pi_{n_{k}})-\frac{1}{n_{k}}\big{(}\varphi^{\Sigma}_{i}(\Pi_{1})+\dots+\varphi^{\Sigma}_{i}(\Pi_{n_{k}})\big{)}\Big{)}\in
W_{n_{k}}.$

Here we denoted by $\varphi_{i}^{\Sigma}$ the composition

$\varphi_{i}^{\Sigma}:(\mathscr{K}^{d})^{\times
m}\xrightarrow{~{}~{}(\varphi_{i})^{\times
m}~{}}\mathbb{R}^{m}\xrightarrow{\langle-,~{}(1,\dots,1)\rangle}\mathbb{R},$

for any integer $m\geq 1$. In the current situation $m=n_{1}\cdots n_{k-1}$.

This completes the construction of the map $\Phi_{k}$. From the induction
hypothesis it follows that $\Phi_{k}$ is indeed
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant.

The crucial property of the map (19) is that it maps a tuple
$(K_{1},\dots,K_{n})\in(\mathscr{K}^{d})^{\times n}$ to the origin if and only
if the tuple satisfies

$\varphi_{i}(K_{1})=\dots=\varphi_{i}(K_{n})$

for each $1\leq i\leq n$.

Now, we return to the proof of the theorem. Assume that for a convex body
$K\in\mathscr{K}^{d}$ there is no solution to Conjecture 1.2 which is of level
$k$ and type $(n_{1},\dots,n_{k})$. We will show that there exists an
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map of the form (18).
Namely, since $K$ does not solve the conjecture, the
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map (19) does not hit the
origin. In other words the map $\Phi_{k}$ restricts to the map

$\Phi_{k}\colon\operatorname{\mathrm{EMP}}_{\mu}(K,n)\longrightarrow
W_{k}({d};{n_{1},\dots,n_{k}})\setminus\\{0\\}.$

(In order to simplify notation we do not introduce new name for the
restriction.) Pre-composing it with the
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant parametrisation map

$C_{k}({d};{n_{1},\dots,n_{k}})\longrightarrow\operatorname{\mathrm{EMP}}_{\mu}(K,n)$

from Proposition 2.8, and post-composing it with the
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant retraction

$W_{k}({d};{n_{1},\dots,n_{k}})\setminus\\{0\\}\longrightarrow
S(W_{k}({d};{n_{1},\dots,n_{k}})),$

yields an $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$C_{k}({d};{n_{1},\dots,n_{k}})\longrightarrow\operatorname{\mathrm{EMP}}_{\mu}(K,n)\longrightarrow
W_{k}({d};{n_{1},\dots,n_{k}})\setminus\\{0\\}\longrightarrow
S(W_{k}({d};{n_{1},\dots,n_{k}}))$

of the form (18), which completes the proof. ∎

##  3\. Proof of Theorem 1.3: Equivariant Obstruction Theory

In this section we give a proof of Theorem 1.3 using equivariant obstruction
theory of tom Dieck [26, Sec. II.3]. Namely, for integers $k\geq 1$, $d\geq 1$
and $n_{1},\dots,n_{k}\geq 2$ we show that an
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$C_{k}({d};{n_{1},\dots,n_{k}})\longrightarrow
S(W_{k}({d};{n_{1},\dots,n_{k}}))$

does not exist if and only if $n_{1},\dots,n_{k}$ are powers of the same prime
number. In order to use the equivariant obstruction theory [26, Sec. II.3], we
need to construct a cellular model $\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$
of the space $C_{k}({d};{n_{1},\dots,n_{k}})$ which is also its
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant deformation retract. Then,
instead of studying the existence of an
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map
$C_{k}({d};{n_{1},\dots,n_{k}})\longrightarrow
S(W_{k}({d};{n_{1},\dots,n_{k}}))$, we focus our attention on the equivalent
problem of the existence of an
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})\longrightarrow
S(W_{k}({d};{n_{1},\dots,n_{k}})).$

###  3.1. Cellular model

Blagojević & Ziegler [7, Sec. 3] have constructed an $(n-1)(d-1)$-dimensional
$\mathfrak{S}_{n}$-equivariant cellular model $\mathscr{F}(d,n)$ of the
Salvetti type for the classical configuration space $F(\mathbb{R}^{d},n)$
which is its $\mathfrak{S}_{n}$-equivariant deformation retract. In this
section, we use this model to construct an
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant model
$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$ of $C_{k}({d};{n_{1},\dots,n_{k}})$
which is of dimension $(d-1)(n_{1}\dots n_{k}-1)$ and is its
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant deformation retract.
Additionally we collect relevant facts about this model needed for an
application of the equivariant obstruction theory.

For integers $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ we simplify
notation on occasions by setting ${\bf n}:=(n_{1},\dots,n_{k})$ and ${\bf
n}^{\prime}:=(n_{1},\dots,n_{k-1})$. However, we will keep the longer notation
in statements of definitions, lemmas, propositions, and theorems.

######  Definition 3.1 (Cellular model).

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. An
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant cell complex
$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$ is defined inductively as follows.

  * –

For $k=1$ set
$\mathscr{C}_{1}({d};{n_{1}}):=\mathscr{F}(\mathbb{R}^{d},n_{1})$ with the
$\mathscr{S}_{1}({n_{1}})\cong\mathfrak{S}_{n_{1}}$-action, consult [7, Sec.
3].

  * –

For $k\geq 2$ set

$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}}):=\mathscr{C}_{k-1}({d};{n_{1},\dots,n_{k-1}})^{\times
n_{k}}\times\mathscr{F}(d,n_{k})$

with the product action of
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})=\mathscr{S}_{k-1}({n_{1},\dots,n_{k-1}})^{\times
n_{k}}\rtimes\mathfrak{S}_{n_{k}}$.

From the recursive formula follows that

$\dim\mathscr{C}_{k}({d};{\bf n})=n_{k}\dim\mathscr{C}_{k-1}({d};{{\bf
n}^{\prime}})+(d-1)(n_{k}-1).$

Since in the base case
$\dim\mathscr{C}_{1}({d};{n})=\dim\mathscr{F}(d,n)=(d-1)(n-1)$, it follows
that the dimension of the complex $\mathscr{C}_{k}({d};{\bf n})$ is indeed

$M_{k}:=(d-1)(n_{1}\dots n_{k}-1).$

Even though $M_{k}$ depends also on parameters $d$ and $n_{1},\dots,n_{k}$,
for simplicity sake they will be omitted from the notation. However, we will
keep parameter $k$ in the notation, as most of the proofs are done using
induction on $k$. This convention will be kept for other notions when there is
not danger of confusion.

######  Proposition 3.2.

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. Then
$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$ is a finite, regular, free
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-CW-complex. Moreover, it is an
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant deformation retract of the
wreath product of configuration spaces $C_{k}({d};{n_{1},\dots,n_{k}})$.

###### Proof.

The proof is done by induction on $k\geq 1$. For the base case $k=1$ let

$r_{1}\colon
F(\mathbb{R}^{d},n)\longrightarrow_{\mathfrak{S}_{n}}\mathscr{F}(\mathbb{R}^{d},n)$

be the $\mathscr{S}_{1}({n_{1}})$-equivariant deformation retraction and
$h_{1}\colon\operatorname{\mathrm{id}}\simeq_{\mathfrak{S}_{n}}i_{1}\circ
r_{1}$ the equivariant homotopy guarantied by [7, Thm. 3.13]. Here
$i_{1}\colon\mathscr{F}(d,n)\hookrightarrow F(\mathbb{R}^{d},n)$ is the
inclusion.

Let now $k\geq 2$. The deformation retraction $r_{k}$ is set to be

$r_{k}:=(r_{k-1})^{\times n_{k}}\times r_{1}\colon C_{k-1}({d};{{\bf
n}^{\prime}})^{\times n_{k}}\times
F(\mathbb{R}^{d},n_{k})\to\mathscr{C}_{k-1}({d};{{\bf n}^{\prime}})^{\times
n_{k}}\times\mathscr{F}(d,n_{k}).$

By the induction hypothesis it follows that the complex is finite, regular and
free, as well as the fact that $r_{k}$ is indeed an $\mathscr{S}_{k}({\bf
n})$-equivariant retraction. Furthermore, the homotopy defined by

$h_{k}:=(h_{k-1})^{\times n_{k}}\times
h_{1}\colon\operatorname{\mathrm{id}}\simeq_{\mathscr{S}_{k}({\bf
n})}i_{k}\circ r_{k}$

makes $r_{k}$ into a deformation retraction. Here,
$i_{k}\colon\mathscr{C}_{l}({d};{\bf n})\hookrightarrow C_{k}({d};{\bf n})$ is
the inclusion. ∎

Let us describe in more detail the cellular structure of the
$\mathfrak{S}_{n}$-CW complex $\mathscr{F}(d,n)$ developed by Blagojević &
Ziegler in [7, Thm. 3.13]. The set of cells of the $\mathfrak{S}_{n}$-CW
complex complex $\mathscr{F}(d,n)$ is

$\\{\check{c}(\sigma,{\bf i}):~{}\sigma\in\mathfrak{S}_{n},~{}{\bf
i}\in[d]^{\times n-1}\\}.$

The dimension if the cell $\check{c}(\sigma,{\bf i})$ is given by the formula
$\dim\check{c}(\sigma,{\bf i})=(i_{1}-1)+\dots+(i_{n-1}-1)$, making the
complex $(n-1)(d-1)$-dimensional. The cellular $\mathfrak{S}_{n}$-action is
given on the cells by $\tau\cdot\check{c}(\sigma,{\bf
i})=\check{c}(\tau\cdot\sigma,{\bf i})$, for each $\tau\in\mathfrak{S}_{n}$.
In particular, from this information, we obtain the following:

  * –

There are in total $n!$ maximal cells, all of which belong to the same
$\mathfrak{S}_{n}$-orbit. Of special importance for later calculations will be
the maximal cell $\check{c}(\operatorname{\mathrm{id}},{\bf d})$, where ${\bf
d}:=(d,\dots,d)\in[d]^{\times n-1}$. Any other maximal cell
$\check{c}(\sigma,{\bf d})$ is equal to
$\sigma\cdot\check{c}(\operatorname{\mathrm{id}},{\bf d})$.

  * –

There are in total $(n-1)n!$ codimension one cells split into $n-1$ orbits of
the group $\mathfrak{S}_{n}$. For $1\leq i\leq n-1$ let us denote by ${\bf
d}_{i}\in[d]^{\times n-1}$ the vector which has value $d$ on all coordinates
different from $i$, and value $d-1$ on coordinate $i$. In this notation,
codimension one cells are encoded by the set

$\\{\check{c}(\sigma,{\bf d}_{i}):~{}\sigma\in\mathfrak{S}_{n},~{}1\leq i\leq
n-1\\}.$

For example, one choice of $n-1$ orbit representatives is
$\check{c}(\operatorname{\mathrm{id}},{\bf
d}_{1}),\dots,\check{c}(\operatorname{\mathrm{id}},{\bf d}_{n-1})$.

Boundary of a maximal cell $\check{c}(\sigma,{\bf d})$ consists of the
following $\binom{n}{1}+\dots+\binom{n}{n-1}$ codimension one cells, as can be
seen either from the boundary relations in [7, Thm. 3.13] or explicitly from
the proof of [7, Lem. 4.1]. For $\sigma\in\mathfrak{S}_{n}$, integer $1\leq
m\leq n-1$, and a $m$-element subset $J\subseteq[n]$, let us denote by
$\sigma_{J}\in\mathfrak{S}_{n}$ the permutation

$\sigma_{J}\colon[n]\longrightarrow[n],\qquad t\longmapsto\sigma(j_{t}),$

where $J=\\{j_{1}<\dots<j_{m})$ and $[n]\setminus J=\\{j_{m+1}<\dots<j_{n}\\}$
is the ordering of the elements. The set of codimension one cells in the
boundary of the maximal cell $\check{c}(\sigma,{\bf d})$ is encoded by the set

$\\{\check{c}(\sigma_{J},{\bf d}_{m}):~{}1\leq m\leq
n-1,~{}J\subseteq[n]\text{ with }|J|=m\\},$ (20)

where two cells $\check{c}(\sigma_{J},{\bf d}_{m})$ and
$\check{c}(\sigma_{I},{\bf d}_{s})$ belong to the same
$\mathfrak{S}_{n}$-orbit if and only if $m=s$. In particular, for each $1\leq
m\leq n-1$, the size of the intersection of the orbit of the cell
$\check{c}(\operatorname{\mathrm{id}},{\bf d}_{m})$ with the above set of
boundary cells is precisely $\binom{n}{m}$.

Extending the notation introduced above, let us define a generalisation of the
special cell $\check{c}(\operatorname{\mathrm{id}},{\bf d})$ of
$\mathscr{F}(d,n)$, which was chosen to be the representative of the orbit of
maximal cells.

######  Definition 3.3 (Orbit representative maximal cell).

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. We define
a maximal cell $e_{k}$ of the cell complex
$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$ inductively as follows.

  * –

For $k=1$ let $e_{1}:=\check{c}(\operatorname{\mathrm{id}},{\bf d})$ be a
maximal cell in $\mathscr{C}_{1}({d};{n})=\mathscr{F}(d,n)$.

  * –

For $k\geq 2$ let us define

$e_{k}:=(e_{k-1})^{\times
n_{k}}\times\check{c}(\operatorname{\mathrm{id}},{\bf d})$

to be a maximal cell in cell complex

$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})=\mathscr{C}_{k-1}({d};{n_{1},\dots,n_{k-1}})^{\times
n_{k}}\times\mathscr{F}(d,n_{k}),$

where $e_{k-1}$ denotes the previously defined maximal cell of
$\mathscr{C}_{k-1}({d};{n_{1},\dots,n_{k-1}})$ and
$\check{c}(\operatorname{\mathrm{id}},{\bf d})$ is a maximal cell of
$\mathscr{F}(d,n_{k})$.

In the next lemma we describe the index set for the codimension one cells
lying in the boundary of the $M_{k}$-cell $e_{k}$.

######  Lemma 3.4.

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers.

  1. (i)

All maximal dimensional cells of $\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$
form a single $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-orbit.

  2. (ii)

The orbit representative maximal cell $e_{k}$ chosen in Definition 3.3 has the
boundary consisting of codimension one cells which are indexed by the set

$B_{k}:=\bigcup_{i=1}^{k}\bigcup_{m=1}^{n_{i}-1}\binom{[n_{i}]}{m}\times[n_{i+1}]\times\dots\times[n_{k}].$

  3. (iii)

The $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-orbit stratification of $B_{k}$ is
given by

$\Big{\\{}\binom{[n_{i}]}{m}\times\\{{\bf j}\\}:~{}1\leq i\leq k,~{}1\leq
m\leq n_{i}-1,~{}{\bf j}\in[n_{i+1}]\times\dots\times[n_{k}]\Big{\\}}.$

###### Proof.

The proof of all three statements is done simultaneously by induction on
$k\geq 1$. The base case $k=1$ of the complex
$\mathscr{C}_{1}({d};{n})=\mathscr{F}(d,n)$ is treated in [7, Lem. 4.1] and
[7, Proof of Lem. 4.2]. See also the boundary description (20). A codimension
one boundary cell $\check{c}(\operatorname{\mathrm{id}}_{J},{\bf
d}_{j})\subseteq\partial\check{c}(\operatorname{\mathrm{id}},{\bf d})$
corresponds to an element

$J\in\bigcup_{m=1}^{n-1}\binom{[n]}{m}=B_{1},$

for each $1\leq j\leq n-1$ and $J\subseteq[n]$ with $|J|=j$.

Let $k\geq 2$. From the inductive definition of the cell complex
$\mathscr{C}_{k}({d};{\bf n})$ and the induction hypothesis it follows that
the $M_{k}$-cells form a single $\mathscr{S}_{k}({\bf n})$-orbit, which
completes the proof of part (i). By the boundary formula of the product
applied to the cell
$e_{k}=(e_{k-1})^{n_{k}}\times\check{c}(\operatorname{\mathrm{id}},{\bf d})$
it follows that the $(M_{k}-1)$-cells in the boundary $\partial e_{k}$ are of
the following two types.

  1. (1)

The first type is

$(e_{k-1})^{i-1}\times\beta\times(e_{k-1})^{n_{k}-i}\times\check{c}(\operatorname{\mathrm{id}},{\bf
d}),$

where $1\leq i\leq n_{k}$ and $\beta$ is the codimension one boundary cell of
$e_{k-1}$. Let $\beta\subseteq\partial e_{k-1}$ be indexed by $b\in B_{k-1}$.
The above $(M_{k}-1)$-cell is set to be indexed by

$(b,i)\in B_{k-1}\times[n_{k}].$

From the inductive definition of the $\mathscr{S}_{k}({\bf n})$-action it
follows that two boundary cells of the first type indexed by

$(b,i),~{}(b^{\prime},i^{\prime})\in B_{k-1}\times[n_{k}]$

are in the same $\mathscr{S}_{k}({\bf n})$-orbit if and only if
$i=i^{\prime}$, and cells $\beta,\beta^{\prime}\subseteq\partial e_{k-1}$ are
in the same $\mathscr{S}_{k-1}({{\bf n}^{\prime}})$-orbit.

  2. (2)

The second type is

$(e_{k-1})^{n_{k}}\times\check{c}(\operatorname{\mathrm{id}}_{J},{\bf d}_{j})$

where $1\leq j\leq n_{k}-1$ and $J\subseteq[n_{k}]$ with $|J|=j$. Here,
$\check{c}(\operatorname{\mathrm{id}}_{J},{\bf d}_{j})$ denotes the
codimension one boundary cell of $\check{c}(\operatorname{\mathrm{id}},{\bf
d})$ described in (20). The above $(M_{k}-1)$-cell is set to be indexed by

$J\in\bigcup_{m=1}^{n_{k}-1}\binom{[n_{k}]}{m}.$

From the definition of the group action it follows that two boundary cells of
the second type, indexed by non-empty proper subsets
$J,J^{\prime}\subseteq[n_{k}]$, are in the same $\mathscr{S}_{k}({\bf
n})$-orbit if and only if $|J|=|J^{\prime}|$.

In particular, we obtained a recursive formula

$B_{k}=B_{k-1}\times[n_{k}]~{}\cup~{}\bigcup_{m=1}^{n_{k}-1}\binom{[n_{k}]}{m},$

which together with the base case description of $B_{1}$ implies the part
(ii).

Since no cell of the first type is in the same $\mathscr{S}_{k}({\bf
n})$-orbit as the cells of the second type, by the above orbit description of
the cells of each of the two types, it follows that the orbit stratification
of $B_{k}$ stated in the part (iii) holds. ∎

###  3.2. Obstructions

Our goal is to show that an $\mathscr{S}_{k}({\bf n})$-equivariant map

$\mathscr{C}_{k}({d};{\bf n})\longrightarrow S(W_{k}({d};{\bf n}))$ (21)

does not exists if and only if $n_{1},\dots,n_{k}$ are all powers of the same
prime number. We have that:

  * –

$\mathscr{C}_{k}({d};{\bf n})$ is a free $\mathscr{S}_{k}({\bf n})$-CW complex
of dimension $M_{k}$,

  * –

the dimension of the $\mathscr{S}_{k}({\bf n})$-sphere $S(W_{k}({d};{\bf n}))$
is $M_{k}-1$, and

  * –

$S(W_{k}({d};{\bf n}))$ is $(M_{k}-1)$-simple and $(M_{k}-2)$-connected.

Applying the equivariant obstruction theory [26, Sec. II.3], we get that the
existence of an $\mathscr{S}_{k}({\bf n})$-equivariant map (21) is equivalent
to the vanishing of the primary obstruction class

${\mathfrak{o}}=[c(f_{k})]\in H^{M_{k}}_{\mathscr{S}_{k}({\bf
n})}(\mathscr{C}_{k}({d};{\bf n});\pi_{M_{k}-1}(S(W_{k}({d};{\bf n})))),$

where $c(f_{k})$ is the obstruction cocycle associated to a general position
equivariant map $f_{k}\colon\mathscr{C}_{k}({d};{\bf n})\longrightarrow
W_{k}({d};{\bf n})$ (see [4, Def. 1.5]). Its values on the $M$-cells $e$ are
given by the following degrees

$c(f_{k})(e)=\deg(r\circ f_{k}\colon\partial e\longrightarrow W_{k}({d};{\bf
n})\setminus\\{0\\}\longrightarrow S(W_{k}({d};{\bf n}))),$

where $r$ is the radial retraction.

The Hurewicz isomorphism [8, Cor. VII.10.8] gives an $\mathscr{S}_{k}({\bf
n})$-module isomorphism

$\pi_{M_{k}-1}(S(W_{k}({d};{\bf n})))\cong H_{M_{k}-1}(S(W_{k}({d};{\bf
n}));\mathbb{Z})=:\mathscr{Z}_{k}.$

The $\mathscr{S}_{k}({\bf n})$-module $\mathscr{Z}_{k}=\langle\xi_{k}\rangle$
is isomorphic to $\mathbb{Z}$ as an abelian group. In order to describe the
$\mathscr{S}_{k}({\bf n})$-module structure on $\mathscr{Z}_{k}$ we first
introduce the notion of “orientation function” as follows.

######  Definition 3.5.

(Orientation function) Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$
be integers. The orientation function

$\operatorname{\mathrm{orient}}\colon\mathscr{S}_{k}({n_{1},\dots,n_{k}})\longrightarrow\\{-1,+1\\}$

is given inductively on $k\geq 1$ as follows.

  * –

For $k=1$ we set $\operatorname{\mathrm{orient}}(\sigma):=({\rm
sgn\,}\sigma)^{d-1}$ for each
$\sigma\in\mathfrak{S}_{n}=\mathscr{S}_{1}({n})$, where ${\rm sgn\,}$ denotes
the sign of the permutation.

  * –

For $k\geq 2$ we set

$\operatorname{\mathrm{orient}}(\Sigma_{1},\dots,\Sigma_{n_{k}};\sigma):=({\rm
sgn\,}\sigma)^{(d-1)\cdot n_{1}\cdots
n_{k-1}}\cdot\operatorname{\mathrm{orient}}(\Sigma_{1})\cdots\operatorname{\mathrm{orient}}(\Sigma_{n_{k}}),$

for any

$(\Sigma_{1},\dots,\Sigma_{n_{k}};\sigma)\in\mathscr{S}_{k-1}({n_{1},\dots,n_{k-1}})^{\times
n_{k}}\rtimes\mathfrak{S}_{n_{k}}=\mathscr{S}_{k}({n_{1},\dots,n_{k}}).$

In the next lemma we show that the action of $\mathscr{S}_{k}({\bf n})$
changes orientation on the vector space $W_{k}({d};{\bf n})$ according to the
orientation function $\operatorname{\mathrm{orient}}$. Consequently, the
$\mathscr{S}_{k}({\bf n})$-module structure on
$\mathscr{Z}_{k}=\langle\xi_{k}\rangle$ is given by

$\Sigma\cdot\xi_{k}=\operatorname{\mathrm{orient}}(\Sigma)\cdot\xi_{k}$ (22)

for each $\Sigma\in\mathscr{S}_{k}({\bf n})$.

######  Lemma 3.6.

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. The group
$\mathscr{S}_{k}({\bf n})$ acts on the vector space $W_{k}({d};{\bf n})$ by
changing the orientation according to the orientation function
$\operatorname{\mathrm{orient}}$ from Definition 3.5. In particular, the map

$\operatorname{\mathrm{orient}}:\mathscr{S}_{k}({n_{1},\dots,n_{k}})\longrightarrow(\\{-1,+1\\},\cdot)$

is a group homomorphism.

###### Proof.

The proof is by induction on $k\geq 1$.

The base case $k=1$ is treated in [7, Sec. 4, p. 69]. Indeed, each
transposition $\tau_{ij}\in\mathfrak{S}_{n}$ acts on $W_{n}$ by reflection in
the hyperplane $x_{i}=x_{j}$, so a permutation $\sigma\in\mathfrak{S}_{n}$
reverses the orientation on $W_{n}$ by ${\rm sgn\,}\sigma$. Thus, $\sigma$
changes the orientation of $W_{n}^{\oplus d-1}$ by $({\rm
sgn\,}\sigma)^{d-1}$.

Let $k\geq 2$. We can split the action of an element

$((\Sigma_{i});\sigma):=(\Sigma_{1},\dots,\Sigma_{n_{k}};\sigma)\in\mathscr{S}_{k-1}({{\bf
n}^{\prime}})^{\times n_{k}}\rtimes\mathfrak{S}_{n_{k}}=\mathscr{S}_{k}({\bf
n})$

on a vector

$((V_{i});v):=(V_{1},\dots,V_{n_{k}};v)\in W_{k-1}({d};{{\bf
n}^{\prime}})^{\oplus n_{k}}\oplus W_{n_{k}}^{\oplus d-1}=W_{k}({d};{\bf n})$

in two steps $(\star)$ and $(\star\star)$ as follows. We have

$((\Sigma_{i});\sigma)\cdot((V_{i});v)=((\Sigma_{i});\operatorname{\mathrm{id}})\cdot((\operatorname{\mathrm{id}});\sigma)\cdot((V_{i});v)\overset{(\star)}{=}((\Sigma_{i});\operatorname{\mathrm{id}})\cdot((V_{\sigma^{-1}(i)});\sigma
v)\overset{(\star\star)}{=}((\Sigma_{i}V_{\sigma^{-1}(i)});\sigma v).$

In the step $(\star)$ permutation $\sigma\in\mathfrak{S}_{n_{k}}$ acts on
$v\in W_{n_{k}}^{\oplus d-1}$ by changing the orientation by $({\rm
sgn\,}\sigma)^{d-1}$, and permutes the $n_{k}$ vectors $V_{i}\in
W_{k-1}({d};{{\bf n}^{\prime}})$ changing the orientation by

$({\rm sgn\,}\sigma)^{\dim W_{k-1}({d};{{\bf n}^{\prime}})}=({\rm
sgn\,}\sigma)^{(d-1)(n_{1}\cdots n_{k-1}-1)}.$

In the step $(\star\star)$ each of the $n_{k}$ elements
$\Sigma_{i}\in\mathscr{S}_{k-1}({{\bf n}^{\prime}})$ acts on
$V_{\sigma^{-1}(i)}\in W_{k-1}({d};{{\bf n}^{\prime}})$ and changes the
orientation by $\operatorname{\mathrm{orient}}(\Sigma_{i})$.

In total, an element
$((\Sigma_{1},\dots,\Sigma_{n_{k}});\sigma)\in\mathscr{S}_{k}({\bf n})$ acts
on the vector space $W_{k}({d};{\bf n})$ and changes the orientation by

$({\rm sgn\,}\sigma)^{d-1}\cdot({\rm sgn\,}\sigma)^{(d-1)(n_{1}\cdots
n_{k-1}-1)}\cdot\operatorname{\mathrm{orient}}(\Sigma_{1})\cdots\operatorname{\mathrm{orient}}(\Sigma_{n_{k}})=\operatorname{\mathrm{orient}}(\Sigma_{1},\dots,\Sigma_{n_{k}};\sigma).$

∎

To evaluate the obstruction cocycle, we use the $\mathscr{S}_{k}({\bf
n})$-equivariant linear projection

$f_{k}\colon W_{k}({d+1};{\bf n})\longrightarrow W_{k}({d};{\bf n}),$

given by forgetting the $d$th coordinate. It serves as a general position map
and is defined inductively as follows.

  * –

For $k=1$ the map $f_{1}\colon W_{n}^{\oplus d}\to W_{n}^{\oplus d-1}$ forgets
the $d$th coordinate.

  * –

For $k\geq 2$ we define $f_{k}$ to be the $\mathscr{S}_{k}({\bf
n})$-equivariant map

$(f_{k-1})^{\oplus n_{k}}\oplus f_{1}:W_{k-1}({d+1};{{\bf
n}^{\prime}})^{\oplus n_{k}}\oplus W_{n_{k}}^{\oplus d}\longrightarrow
W_{k-1}({d};{{\bf n}^{\prime}})^{\oplus n_{k}}\oplus W_{n_{k}}^{\oplus d-1}.$

Since by construction we have
$\mathscr{C}_{1}({d};{n})=\mathscr{F}(d,n)\subseteq W_{n}^{\oplus
d}=W_{1}({d+1};{n})$ (see [7, Sec. 3]), by restriction of the domain we can
speak of the $\mathscr{S}_{1}({n})$-equivariant map

$f_{1}\colon\mathscr{C}_{1}({d};{n})\longrightarrow W_{1}({d};{n}).$

By induction on $k\geq 1$ it follows that $\mathscr{C}_{k}({d};{\bf
n})\subseteq W_{k}({d+1};{\bf n})$, so we may speak about the
$\mathscr{S}_{k}({\bf n})$-equivariant map

$f_{k}\colon\mathscr{C}_{k}({d};{\bf n})\longrightarrow W_{k}({d};{\bf n}).$

######  Lemma 3.7.

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. Then the
following statements hold.

  1. (i)

The linear map $f_{k}$ maps all $M_{k}$-cells of
$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$ by a cellular homeomorphism to the
same star-shaped $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-neighbourhood
$\mathscr{B}_{k}\subseteq W_{k}({d};{n_{1},\dots,n_{k}})$ of the origin.

  2. (ii)

There exists an orientation of the $M_{k}$\- and $(M_{k}-1)$-cells of the cell
complex $\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$ such that the cellular
action of $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$ on $M_{k}$\- and
$(M_{k}-1)$-cells changes the orientation according to the orientation
function $\operatorname{\mathrm{orient}}$ defined in Definition 3.5.

  3. (iii)

Assuming the orientations of the cells of
$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$ from part (ii), the obstruction
cocycle $c(f_{k})$ has the value $+1$ on all oriented $M_{k}$-cells of
$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$.

  4. (iv)

Assuming the orientations of the cells of
$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$ from part (ii), the following
formula of cellular chains holds

$\partial e_{k}=\sum_{b\in B_{k}}{\rm sgn\,}_{k}(b)\cdot e(b)\in
C_{M_{k}-1}(\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})),$

where $e(b)$ denotes the $(M_{k}-1)$-cell in the boundary of $e_{k}$ indexed
by $b\in B_{k}$ in light of Lemma 3.4(ii) , and ${\rm
sgn\,}_{k}(b)\in\\{-1,+1\\}$ is a sign function constant on each
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-orbit.

###### Proof.

As usual, all statements are proved by the (same) induction on $k\geq 1$. The
base case $k=1$ is treated in [7, Lem. 4.1]. There, the $(M_{1}-1)$-cells in
$\partial\mathscr{B}_{1}$ are oriented such that they all appear with a sign
$+1$ in the cellular boundary formula of $\mathscr{B}_{1}$.

Now, assume that $k\geq 2$.

  * –

We define $\mathscr{B}_{k}:=(\mathscr{B}_{k-1})^{\times
n_{k}}\times\mathscr{B}_{1}$, where $\mathscr{B}_{k-1}\subseteq
W_{k-1}({d};{{\bf n}^{\prime}})$ is the star-shaped neighbourhood from the
induction hypothesis. Any $M_{k}$-cell $e$ of $\mathscr{C}_{k}({d};{\bf n})$
equals to the product of $n_{k}$ maximal cells of $\mathscr{C}_{k-1}({d};{{\bf
n}^{\prime}})$ and a maximal cell of $\mathscr{F}(d,n_{k})$. Therefore, by
induction hypothesis and the inductive definition of the map $f_{k}$, it
follows that $f_{k}$ restricted to the cell $e$ is cellular homeomorphism.

  * –

Let $\mathscr{B}_{k}$ and the $(M_{k}-1)$-cells in $\partial\mathscr{B}_{k}$
be endowed with the product orientation. The map $f_{k}$ sends each $M_{k}$\-
and $(M_{k}-1)$-cell of $\mathscr{C}_{k}({d};{\bf n})$ homeomorphically to
$\mathscr{B}_{k}$ and a $(M_{k}-1)$-cell in the boundary of $\mathscr{B}_{k}$,
respectively. Thus, we can set orientation on each $M_{k}$\- and
$(M_{k}-1)$-cell of $\mathscr{C}_{k}({d};{\bf n})$ such that $f_{k}$
restricted to them is orientation preserving. By Lemma 3.6 the
$\mathscr{S}_{k}({\bf n})$-action on $\mathscr{B}_{k}$ changes the orientation
by the sign given by orientation function $\operatorname{\mathrm{orient}}$, so
the same is true on the cells of $\mathscr{C}_{k}({d};{\bf n})$.

  * –

Since $f_{k}$ is orientation preserving cellular homeomorphism, we have

$c(f_{k})(e)=\deg(r\circ f_{k}\colon\partial e\longrightarrow W_{k}({d};{\bf
n})\setminus\\{0\\}\longrightarrow\partial\mathscr{B}_{k})=1,$

for each $M_{k}$-cell $e\subseteq\mathscr{C}_{k}({d};{\bf n})$.

  * –

Since $\mathscr{C}_{k}({d};{\bf n})$ is regular, we need to show only that
${\rm sgn\,}(b)={\rm sgn\,}(b^{\prime})$ for any $b,b^{\prime}\in B_{k}$ in
the same orbit. By the boundary of the cross-product formula for
$e_{k}=(e_{k-1})^{\times n_{k}}\times\check{c}(\operatorname{\mathrm{id}},{\bf
d})$, we have the following equality of cellular $(M_{k}-1)$-chains:

$\partial
e_{k}=\sum_{i=1}^{n_{k}}(-1)^{(i-1)M_{k-1}}e_{k-1}^{i-1}\times\partial
e_{k-1}\times e_{k-1}^{n_{k}-i}\times\check{c}(\operatorname{\mathrm{id}},{\bf
d})+(-1)^{n_{k}M_{k-1}}e_{k-1}^{n_{k}}\times\partial\check{c}(\operatorname{\mathrm{id}},{\bf
d}).$

By the induction hypothesis, we have

$\partial e_{k-1}=\sum_{c\in B_{k-1}}{\rm sgn\,}_{k-1}(c)\cdot
e(c)\qquad\text{ and }\qquad\partial\check{c}(\operatorname{\mathrm{id}},{\bf
d})=\sum_{a\in B_{1}}{\rm sgn\,}_{1}(a)\check{c}(a).$

Moreover, we have ${\rm sgn\,}_{k-1}(c)={\rm sgn\,}_{k-1}(c^{\prime})$ if
$c,c^{\prime}\in B_{k-1}$ are in the same $\mathscr{S}_{k-1}({{\bf
n}^{\prime}})$-orbit and ${\rm sgn\,}_{1}(a)={\rm sgn\,}_{1}(a^{\prime})$ if
$a,a^{\prime}\in B_{1}$ are in the same $\mathfrak{S}_{n_{k}}$-orbit.
Similarly to the proof of Lemma 3.4, boundary cells of $e_{k}$ are divided
into two types.

    1. (1)

Cells of the first type are of the form

$(e_{k-1})^{\times n_{k}}\times e(c)\times(e_{k-1})^{\times
n_{k}-i}\times\check{c}(\operatorname{\mathrm{id}},{\bf d})$

for $1\leq i\leq n_{k}$ and $c\in B_{k-1}$. By part (iii) of Lemma 3.4 or its
proof, it is seen that such a cell receives a label

$(c,i)\in B_{k-1}\times[n_{k}]\subseteq B_{k}.$

Moreover, two cells labeled by $(c,i),(c^{\prime},i^{\prime})\in
B_{k-1}\times[n_{k}]\subseteq B_{k}$ are in the same $\mathscr{S}_{k}({\bf
n})$-orbit if and only if $i=i^{\prime}$ and $c,c^{\prime}\in B_{k-1}$ are in
the same $\mathscr{S}_{k-1}({{\bf n}^{\prime}})$-orbit. Therefore, by the
inductive sign formula for the cells of the first type

${\rm sgn\,}_{k}(c,i)=(-1)^{(i-1)M_{k-1}}\cdot{\rm sgn\,}_{k-1}(c)$

the claim follows for the boundary cells of the first type.

    2. (2)

Cells of the second type are of the form

$(e_{k-1})^{\times n_{k}}\times\check{c}(a)$

for $a\in B_{1}$. Again by part (iii) of Lemma 3.4 or its proof, it is seen
that such a cell receives a label $a\in B_{1}\subseteq B_{k}$. Moreover, two
cells labeled by $a,a^{\prime}\in B_{1}\subseteq B_{k}$ are in the same
$\mathscr{S}_{k}({\bf n})$-orbit if and only if $a,a^{\prime}\in B_{1}$ are in
the same $\mathfrak{S}_{n_{k}}$-orbit. Therefore, by the inductive sign
formula for the cells of the second type

${\rm sgn\,}_{k}(a)=(-1)^{n_{k}M_{k-1}}\cdot{\rm sgn\,}_{1}(a)$

the claim follows for the boundary cells of the second type.

Finally, by part (iii) of Lemma 3.4 or its proof, no cell of the first type is
in the same orbit as a cell of the second type, so the proof is completed.

∎

###  3.3. When is the obstruction cocycle a coboundary?

In this section we discuss when the computed cocycle $c(f_{k})$ is a
coboundary, and consequently complete the proof of Theorem 1.3. The
orientation of cells of the complex $\mathscr{C}_{k}({d};{\bf n})$ is
understood to be the one from Lemma 3.7(ii). The following lemma is a
generalisation of the $k=1$ case treated in [7, Lem. 4.2].

######  Lemma 3.8.

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. Then the
following statements hold.

  1. (i)

The value of the coboundary $\delta w$ of any equivariant cellular cochain

$w\in
C^{M_{k}-1}_{\mathscr{S}_{k}({n_{1},\dots,n_{k}})}(\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}});\mathscr{Z}_{k})$

is the same on each $M_{k}$-cell of $\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$
and is equal to the $\mathbb{Z}$-linear combination of binomial coefficients

$\binom{n_{1}}{1},\dots,\binom{n_{1}}{n_{1}-1},\dots,\binom{n_{k}}{1},\dots,\binom{n_{k}}{n_{k}-1}.$
(23)

  2. (ii)

For any $\mathbb{Z}$-linear combination of binomial coefficients (23), there
exists a cochain whose coboundary takes precisely that value on all
$M_{k}$-cells of $\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})$.

###### Proof.

Let

$w\in C^{M_{k}-1}_{\mathscr{S}_{k}({\bf n})}(\mathscr{C}_{k}({d};{\bf
n});\mathscr{Z}_{k})=\hom_{\mathscr{S}_{k}({\bf
n})}(C_{M_{k}-1}(\mathscr{C}_{k}({d};{\bf n})),\mathscr{Z}_{k})$

be a cellular cochain. The group $\mathscr{S}_{k}({\bf n})$:

  * –

changes the orientation of any $M_{k}$\- or $(M_{k}-1)$-cell according to the
orientation function $\operatorname{\mathrm{orient}}$ (consult Lemma 3.7
(ii)), and

  * –

acts on $\mathscr{Z}_{k}$ by multiplication with the value of
$\operatorname{\mathrm{orient}}$ (see (22)).

Therefore, an $\mathscr{S}_{k}({\bf n})$-module morphism of the form

$C_{M_{k}-1}(\mathscr{C}_{k}({d};{\bf
n}))\longrightarrow\mathscr{Z}_{k}~{}~{}~{}\text{
or}~{}~{}~{}C_{M_{k}}(\mathscr{C}_{k}({d};{\bf
n}))\longrightarrow\mathscr{Z}_{k}$

is constant on each $\mathscr{S}_{k}({\bf n})$-orbit.

In particular, since $M_{k}$-cells form a single $\mathscr{S}_{k}({\bf
n})$-orbit by Lemma 3.4(i), it follows that the coboundary $\delta w$ has the
same value on all $M_{k}$-cells. Therefore, we restrict our attention to the
orbit representative $M_{k}$-cell $e_{k}$ introduced in Definition 3.3 and the
value $(\delta w)(e_{k})=w(\partial e_{k})$.

From Lemma 3.7(iv) we have that

$\partial e_{k}=\sum_{b\in B_{k}}{\rm sgn\,}(b)\cdot e(b)\in
C_{M_{k}-1}(\mathscr{C}_{k}({d};{\bf n})),$

with the sign function ${\rm sgn\,}$ being constant on $\mathscr{S}_{k}({\bf
n})$-orbits of $(M_{k}-1)$-cells. According to Lemma 3.4(iii), the
$\mathscr{S}_{k}({\bf n})$-orbit stratification of the index set $B_{k}$ is
given by

$\Big{\\{}\binom{[n_{i}]}{m}\times\\{{\bf j}\\}:~{}1\leq i\leq k,~{}1\leq
m\leq n_{i}-1,~{}{\bf j}\in[n_{i+1}]\times\dots\times[n_{k}]\Big{\\}},$

and we may denote by $w(i,m;{\bf j})$ and ${\rm sgn\,}(i,m;{\bf j})$ the
values of $w$ and ${\rm sgn\,}$ on the orbit containing the corresponding
stratum. With this notation in hand, we have

$(\delta w)(e_{k})=w(\partial e_{k})=\sum_{1\leq i\leq k}~{}\sum_{1\leq m\leq
n_{i}-1}~{}\sum_{{\bf
j}\in[n_{i+1}]\times\dots\times[n_{k}]}\binom{n_{i}}{m}{\rm sgn\,}(i,m;{\bf
j})w(i,m;{\bf j}),$ (24)

which proves part (i) of the lemma.

To prove part (ii), let us show that any $\mathbb{Z}$-linear combination

$\sum_{1\leq i\leq k}~{}\sum_{1\leq m\leq
n_{i}-1}x_{i,m}\cdot\binom{n_{i}}{m}$

is equal to the value $(\delta w)(e_{k})$, for some cellular
$(M_{k}-1)$-cochain $w$. The value of $w$ can be chosen independently on each
orbit. For example, one can set $w$ to be the equivariant extension of the map
given on boundary cells of $e_{k}$ as

$w(e(b)):=\begin{cases}x_{i,m}\cdot{\rm
sgn\,}(i,m;(1,\dots,1)),&b\in\binom{[n_{i}]}{m}\times\\{(1,\dots,1)\\}\text{
for some }i\in[k],m\in[n_{i}-1],\\\ 0,&\text{otherwise},\end{cases}$

for each $b\in B_{k}$. Indeed, by (24) we have

$(\delta w)(e_{k})=\sum_{1\leq i\leq k}~{}\sum_{1\leq m\leq
n_{i}-1}\binom{n_{i}}{m}{\rm sgn\,}(i,m;{\bf 1})w(i,m;{\bf 1})=\sum_{1\leq
i\leq k}~{}\sum_{1\leq m\leq n_{i}-1}\binom{n_{i}}{m}x_{i,m},$

where ${\bf 1}:=(1,\dots,1)\in[n_{i+1}]\times\dots\times[n_{k}]$ is the
notation in each summand. ∎

The next elementary consequence of the Ram’s result [24] is used in the proof
of Theorem 1.3.

######  Lemma 3.9.

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. Then we
have that

$\gcd\Big{\\{}\binom{n_{i}}{1},\dots,\binom{n_{i}}{n_{i}-1}:~{}1\leq i\leq
k\Big{\\}}=\begin{cases}p,&\text{if all }n_{i}\text{ are powers of the same
prime }p,\\\ 1,&\text{otherwise.}\end{cases}$

###### Proof.

By the work of Ram’s result [24] it follows that for each $1\leq i\leq k$ we
have

$N_{i}:=\gcd\Big{\\{}\binom{n_{i}}{1},\dots,\binom{n_{i}}{n_{i}-1}\Big{\\}}=\begin{cases}p,&\text{if
}n_{i}\text{ is a power of a prime }p,\\\ 1,&\text{otherwise.}\end{cases}$

Therefore, we have $\gcd(N_{1},\dots,N_{k})\neq 1$ if and only if $n_{1},\dots
n_{k}$ are all powers of the same prime $p$, in which case we have
$\gcd(N_{1},\dots,N_{k})=p$. ∎

Finally, we are ready to give a proof of Theorem 1.3.

###### Proof of Theorem 1.3.

By [26, Sec. II.3] the equivariant map exists if and only if the cohomology
class ${\mathfrak{o}}=[c(f_{k})]$ vanishes. This happens if and only if
$c(f_{k})$ is the coboundary of some equivariant cellular cochain in
$C^{M-1}_{\mathscr{S}_{k}({\bf n})}(\mathscr{C}_{k}({d};{\bf
n});\mathscr{Z}_{k})$. By Lemmas 3.7 and 3.8 this happens if and only if there
exists a linear combination of binomial coefficients

$\binom{n_{1}}{1},\dots,\binom{n_{1}}{n_{1}-1},\dots,\binom{n_{k}}{1},\dots,\binom{n_{k}}{n_{k}-1}$

which is equal to $1$. This equivalent to the fact that the greatest common
divisor of all such binomial coefficients equals $1$, which by Lemma 3.9
happens if and only if $n_{1},\dots,n_{k}$ are not all powers of the same
prime number. ∎

##  4\. Existence in Theorem 1.3: little cubes operad

In this section we give a short proof of the existence part of Theorem 1.3
using the little cubes operad structural map. As before, let us set ${\bf
n}:=(n_{1},\dots,n_{k})$. Our goal is to show the existence of an
$\mathscr{S}_{k}({\bf n})$-equivariant map

$C_{k}({d};{\bf n})\longrightarrow S(W_{k}({d};{\bf n}))$

in the case when $n_{1},\dots,n_{k}$ are not all powers of the same prime
number.

In the remainder of the section, we will consider $\mathscr{S}_{k}({\bf n})$
as the subgroup of the symmetric group $\mathfrak{S}_{n}$, where
$n:=n_{1}\cdots n_{k}$. Indeed, $\mathscr{S}_{k}({\bf n})$ is the automorphism
group of the poset $P_{k}({\bf n})$ by Lemma 2.5. In particular, it permutes
the $n$ elements of level $k$ of $P_{k}({\bf n})$. Hence, it is a subgroup of
all permutations of the level $k$ elements of $P_{k}({\bf n})$, which can be
identified with $\mathfrak{S}_{n}$. Note that this description specifies the
inclusion of $\mathscr{S}_{k}({\bf n})$ into $\mathfrak{S}_{n}$.

Let $(\mathrm{Cubes}_{d}({n}))_{n\geq 1}$ denote the little cubes operad as
introduced by May [18]. See also [3, Sec. 7.3] for notation. Each
$\mathrm{Cubes}_{d}({n})$ is a free $\mathfrak{S}_{n}$-space. The operad comes
with the structural map

$\mu\colon(\mathrm{Cubes}_{d}({m_{1}})\times\dots\times\mathrm{Cubes}_{d}({m_{k}}))\times\mathrm{Cubes}_{d}({k})\longrightarrow\mathrm{Cubes}_{d}({m_{1}+\dots+m_{k}}),$

for any integers $k\geq 1$ and $m_{1},\dots,m_{k}\geq 1$. Moreover, we have
the following fact [18, Thm. 4.8].

######  Lemma 4.1.

Let $d\geq 1$ and $n\geq 1$ be integers. Then, $\mathrm{Cubes}_{d}({n})$ is
$\mathfrak{S}_{n}$-homotopy equivalent to the configuration space
$F(\mathbb{R}^{d},n)$.

The special case of the structural map when $m_{1}=\dots=m_{k}$ is useful for
us, as explained in the following lemma, consult also [3, Lem. 7.2].

######  Lemma 4.2.

Let $d\geq 1$ and $n\geq 1$ be integers. The structural map

$\mu\colon\mathrm{Cubes}_{d}({m})^{\times
k}\times\mathrm{Cubes}_{d}({k})\longrightarrow\mathrm{Cubes}_{d}({mk})$

is $((\mathfrak{S}_{m})^{\times k}\rtimes\mathfrak{S}_{k})$-equivariant. The
action of $(\mathfrak{S}_{m})^{\times k}\rtimes\mathfrak{S}_{k}$ on the domain
is assumed to be the wreath product action from Definition 2.6, and the action
on the codomain is given by restriction $(\mathfrak{S}_{m})^{\times
k}\rtimes\mathfrak{S}_{k}=\mathscr{S}_{2}({m,k})\subseteq\mathfrak{S}_{mk}$.

Now, we prove the following auxiliar result.

######  Lemma 4.3.

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. There
exists an $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$C_{k}({d};{n_{1},\dots,n_{k}})\longrightarrow F(\mathbb{R}^{d},n),$

where $n:=n_{1}\cdots n_{k}$ and the action on the codomain is the restriction
action $\mathscr{S}_{k}({n_{1},\dots,n_{k}})\subseteq\mathfrak{S}_{n}$.

###### Proof.

We will show the existence of an
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$\gamma_{k}\colon C_{k}({d};{n_{1},\dots,n_{k}})\longrightarrow
F(\mathbb{R}^{d},n),$

by induction on $k\geq 1$. The base case $k=1$ the map $\gamma_{1}$ is the
equality.

Assume $k\geq 2$. As before, let $n^{\prime}:=n_{1}\cdots n_{k-1}$ and ${\bf
n}^{\prime}:=(n_{1},\dots,n_{k-1})$. The map $\gamma_{k}$ is defined by the
following diagram:

${C_{k-1}(d;{\bf n}^{\prime})^{\times n_{k}}\times
F(\mathbb{R}^{d},n_{k})}$${F(\mathbb{R}^{d},n^{\prime})^{\times n_{k}}\times
F(\mathbb{R}^{d},n_{k})}$${F(\mathbb{R}^{d},n)}$${\mathrm{Cubes}_{d}({n^{\prime}})^{\times
n_{k}}\times\mathrm{Cubes}_{d}({n_{k}})}$${\mathrm{Cubes}_{d}({n}).}$$\scriptstyle{(\gamma_{k-1})^{\times
n_{k}}\times\operatorname{\mathrm{id}}}$$\scriptstyle{\simeq}$$\scriptstyle{\mu}$$\scriptstyle{\simeq}$

The group $\mathscr{S}_{k}({\bf n})$ acts on
$F(\mathbb{R}^{d},n^{\prime})^{\times n_{k}}\times F(\mathbb{R}^{d},n_{k})$ by
the wreath product action induced from $\mathscr{S}_{k-1}({{\bf
n}^{\prime}})\subseteq\mathfrak{S}_{n^{\prime}}$, so the top horizontal map is
$\mathscr{S}_{k}({\bf n})$-equivariant by the induction hypothesis. Both
vertical maps are $\mathscr{S}_{k}({\bf n})$-equivariant by Lemma 4.1. Indeed,
the left one is the product of equivariant homotopy equivalences, hence its
homotopy inverse is equivariant as well. Finally, the bottom horizontal map is
$\mathscr{S}_{k}({\bf n})$-equivariant by Lemma 4.2. ∎

Next, we prove an additional auxiliar result.

######  Lemma 4.4.

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. Then,
there exists an $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-module monomorphism

${W_{n}^{\oplus d-1}}$${W_{k}({d};{n_{1},\dots,n_{k}}),}$

where $n:=n_{1}\cdots n_{k}$ and the action on the codomain is the restriction
action $\mathscr{S}_{k}({n_{1},\dots,n_{k}})\subseteq\mathfrak{S}_{n}$.

###### Proof.

Since we have $W_{k}({d};{\bf n})\cong_{\mathscr{S}_{k}({\bf
n})}W_{k}({1};{\bf n})^{\oplus d-1}$ it is enough to show there exists an
$\mathscr{S}_{k}({\bf n})$-equivariant monomorphism

${\mu_{k}:W_{n}}$${~{}W_{k}({1};{\bf n}).}$

We will prove this by induction on $k\geq 1$. In the base case $k=1$ we set
$\mu_{1}$ to be the identity.

Assume $k\geq 2$. Let ${\bf n}^{\prime}:=(n_{1},\dots,n_{k-1})$ and
$n^{\prime}=n_{1}\cdots n_{k-1}$. It is enough to show there exists an
$\mathscr{S}_{k}({\bf n})$-equivariant monomorphism of the form

${W_{n}}$${~{}(W_{n^{\prime}})^{\oplus n_{k}}\oplus W_{n_{k}},}$ (25)

where $\mathscr{S}_{k}({\bf n})=\mathscr{S}_{k-1}({{\bf
n}^{\prime}})^{n_{k}}\rtimes\mathfrak{S}_{n_{k}}$ acts on the codomain by the
wreath product action from Definition 2.6 and $\mathscr{S}_{k-1}({{\bf
n}^{\prime}})$ acts on $W_{n^{\prime}}$ by the restriction
$\mathscr{S}_{k-1}({{\bf n}^{\prime}})\subseteq\mathfrak{S}_{n^{\prime}}$.
Indeed, then the desired map can be set to be the $\mathscr{S}_{k}({\bf
n})$-equivariant composition

${W_{n}~{}}$${~{}W_{n^{\prime}}^{\oplus n_{k}}\oplus
W_{n_{k}}~{}}$${~{}W_{k-1}({1};{{\bf n}^{\prime}})^{\oplus n_{k}}\oplus
W_{n_{k}}~{}=~{}W_{k}({1};{{\bf n}}).}$$\scriptstyle{\eqref{eq: inductive step
map for W}}$$\scriptstyle{\mu_{k-1}^{\oplus
n_{k}}\oplus\operatorname{\mathrm{id}}}$

Finally, the map (25) can be constructed as follows. Let

$w=(w_{1},\dots w_{n_{k}})\in
W_{n}\subseteq(\mathbb{R}^{n^{\prime}})^{n_{k}},$

where $w_{1}+\dots+w_{n_{k}}=0$. Furthermore, let us denote by

  * –

$\overline{w}_{i}$ the average of the $n^{\prime}$ coordinates of the vector
$w_{i}\in\mathbb{R}^{n^{\prime}}$ for each $1\leq i\leq n_{k}$, and

  * –

$\overline{w}$ the average of the $n_{k}$ coordinates of the vector
$(\overline{w}_{1},\dots,\overline{w}_{n_{k}})\in\mathbb{R}^{n_{k}}$.

With this notation, we have that

$w_{i}-\overline{w}_{i}\cdot(1,\dots,1)\in W_{n^{\prime}}$

for each $1\leq i\leq n_{k}$, as well as

$(\overline{w}_{1},\dots,\overline{w}_{n_{k}})-\overline{w}\cdot(1,\dots,1)\in
W_{n_{k}}.$

Finally, we set the map (25) to be such that it sends $w\in W_{n}$ to the
vector

$\Big{(}w_{i}-\overline{w}_{i}\cdot(1,\dots,1)\Big{)}_{i=1}^{n_{k}}\oplus\Big{(}(\overline{w}_{1},\dots,\overline{w}_{n_{k}})-\overline{w}\cdot(1,\dots,1)\Big{)}\in(W_{n^{\prime}})^{\oplus
n_{k}}\oplus W_{n_{k}}.$

This map is injective. Moreover, the restriction action $\mathscr{S}_{k}({\bf
n})\subseteq\mathfrak{S}_{n}$ on $W_{n}$ and the wreath product action of
$\mathscr{S}_{k-1}({{\bf n}^{\prime}})^{n_{k}}\rtimes\mathfrak{S}_{n_{k}}$ on
$(W_{n^{\prime}})^{\oplus n_{k}}\oplus W_{n_{k}}$ make it
$\mathscr{S}_{k}({\bf n})$-equivariant. ∎

Finally, we give the second proof of the existence part of Theorem 1.3.

######  Corollary 4.5 (Existence in Thm. 1.3).

Let $k\geq 1$, $d\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. Assume
that $n_{1},\dots,n_{k}$ are not all powers of the same prime number. Then,
there exists an $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$C_{k}({d};{n_{1},\dots,n_{k}})~{}\longrightarrow~{}S(W_{k}({d};{n_{1},\dots,n_{k}})).$

###### Proof.

If $n_{1},\dots,n_{k}$ are not all powers of the same prime number, in
particular $n:=n_{1}\cdots n_{k}$ is not a power of a prime number, hence by
[7, Thm. 1.2] there exists an $\mathfrak{S}_{n}$-equivariant map

$F(\mathbb{R}^{d},n)~{}\longrightarrow~{}W_{n}^{\oplus d-1}\setminus\\{0\\}.$

Let us set ${\bf n}:=(n_{1},\dots,n_{k})$. Precomposing this map with the map
from Lemma 4.3 and postcomposing with the monomorphism from Lemma 4.4 we get
an $\mathscr{S}_{k}({\bf n})$-equivariant composition

${C_{k}(d;{\bf n})~{}}$${~{}F(\mathbb{R}^{d},n)~{}}$${~{}W_{n}^{\oplus
d-1}\setminus\\{0\\}~{}}$${~{}W_{k}(d;{\bf n})\setminus\\{0\\}}$

where we consider $\mathscr{S}_{k}({\bf n})\subseteq\mathfrak{S}_{n}$. ∎

##  5\. Existence in Theorem 1.3: the Özaydin trick

In this section we give yet another short proof of the existence of an
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})~{}\longrightarrow~{}S(W_{k}({d};{n_{1},\dots,n_{k}}))$
(26)

when $n_{1},\dots,n_{k}$ are not all powers of the same prime number. We rely
on the equivariant obstruction theory, as presented by tom Dieck [26, Sec.
II.3], and appropriate a trick developed by Özaydin in [22].

Let ${\bf n}:=(n_{1},\dots,n_{k})$ and $M_{k}:=(d-1)(n_{1}\cdots n_{k}-1)$ as
before. Not that

  * –

$\mathscr{C}_{k}({d};{\bf n})$ is a free $\mathscr{S}_{k}({\bf n})$-cell
complex of dimension $M_{k}$, and

  * –

the $\mathscr{S}_{k}({\bf n})$-sphere $S(W_{k}({d};{\bf n}))$ is
$(M_{k}-1)$-simple and $(M_{k}-2)$-connected.

Consequently, an $\mathscr{S}_{k}({\bf n})$-equivariant map (27) exists if and
only if the primary obstruction

$\mathfrak{o}_{\mathscr{S}_{k}({\bf n})}\in H^{M_{k}}_{\mathscr{S}_{k}({\bf
n})}(\mathscr{C}_{k}({d};{\bf n});\mathscr{Z}_{k})$

vanishes. Here,as before, $\mathscr{Z}_{k}:=\pi_{M_{k}-1}(S(W_{k}({d};{\bf
n})))$ denotes the coefficient module.

In same way, for any subgroup $G\subseteq\mathscr{S}_{k}({\bf n})$, the
existence of a $G$-equivariant map

$\mathscr{C}_{k}({d};{n_{1},\dots,n_{k}})~{}\longrightarrow~{}S(W_{k}({d};{n_{1},\dots,n_{k}}))$
(27)

is equivalent to the vanishing of the obstruction class

$\mathfrak{o}_{G}\in H^{M_{k}}_{G}(\mathscr{C}_{k}({d};{\bf
n});\mathscr{Z}_{k}),$

which in particular is the restriction of the class
$\mathfrak{o}_{\mathscr{S}_{k}({\bf n})}$.

For each integer $n\geq 2$ and a prime number $p$, let us fix a $p$-Sylow
subgroup $\mathfrak{S}_{n}^{(p)}\subseteq\mathfrak{S}_{n}$. For more details
on Sylow subgroups, see for example [11, Sec. 4.5].

######  Definition 5.1.

Let $k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. For a prime number
$p$, let us define a subgroup
$\mathscr{S}_{k}({n_{1},\dots,n_{k}})^{(p)}\subseteq\mathscr{S}_{k}({n_{1},\dots,n_{k}})$
inductively as follows.

  1. (i)

For $k=1$ we set $\mathscr{S}_{1}({n}):=\mathfrak{S}_{n}^{(p)}$.

  2. (ii)

Assume $k\geq 2$. Then, we define

$\mathscr{S}_{k}({n_{1},\dots,n_{k}})^{(p)}:=(\mathscr{S}_{k-1}({n_{1},\dots,n_{k-1}})^{(p)})^{\times
n_{k}}\rtimes\mathfrak{S}_{n_{k}}^{(p)}\subseteq\mathscr{S}_{k-1}({n_{1},\dots,n_{k-1}})^{\times
n_{k}}\rtimes\mathfrak{S}_{n_{k}}.$

Notice that $\mathscr{S}_{k}({\bf n})^{(p)}\subseteq\mathscr{S}_{k}({\bf n})$
is indeed a $p$-Sylow subgroup. For $k=1$ this follows from definition, while
for $k\geq 2$ the recursive formula

$\big{|}\mathscr{S}_{k}({\bf n})^{(p)}\big{|}=\big{|}\mathscr{S}_{k-1}({{\bf
n}^{\prime}})^{(p)}\big{|}^{n_{k}}\cdot\big{|}\mathfrak{S}_{n_{k}}^{(p)}\big{|}$

implies that the order $|\mathscr{S}_{k}({\bf n})^{(p)}|$ is the maximal power
of $p$ which divides the order of $|\mathscr{S}_{k}({n_{1},\dots,n_{k}})|$.

######  Lemma 5.2.

Let $d\geq 1$, $k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers and let $p$
be a prime number. If $n_{1},\dots,n_{k}$ are not all powers of $p$, then the
obstruction class

$\mathfrak{o}_{\mathscr{S}_{k}({\bf n})^{(p)}}\in
H^{M_{k}}_{\mathscr{S}_{k}({\bf n})^{(p)}}(\mathscr{C}_{k}({d};{\bf
n});\mathscr{Z}_{k})$

vanishes.

###### Proof.

Assume $n_{1},\dots,n_{k}$ are not all powers of a prime $p$. Let us show that
the action of the $p$-Sylow subgroup $\mathscr{S}_{k}({\bf
n})^{(p)}\subseteq\mathscr{S}_{k}({\bf n})$ on the sphere $S(W_{k}({d};{\bf
n}))$ has a fixed point. This would produce a (constant) $\mathscr{S}_{k}({\bf
n})^{(p)}$-equivariant map

$\mathscr{C}_{k}({d};{\bf n})\longrightarrow S(W_{k}({d};{\bf n})).$

Hence, from the discussion at the beginning of this section, we would have
that $\mathfrak{o}_{\mathscr{S}_{k}({\bf n})^{(p)}}=0$.

Since

$S(W_{k}({d};{\bf n}))\subseteq W_{k}({d};{\bf n})\setminus\\{0\\}$

is an $\mathscr{S}_{k}({\bf n})$-equivariant deformation retract, it is enough
to show that $\mathscr{S}_{k}({\bf n})^{(p)}\subseteq\mathscr{S}_{k}({\bf n})$
has a nonzero fixed point in the vector space $W_{k}({d};{\bf n})$. We prove
this fact by induction on $k\geq 1$.

  1. (i)

If $k=1$, then $n:=n_{1}$ is not a power of $p$. Thus, the $p$-Sylow subgroup
$\mathfrak{S}_{n}^{(p)}\subseteq\mathfrak{S}_{n}$ does not act transitively on
the set $\\{1,\dots,n\\}$. Consequently, $\mathfrak{S}_{n}^{(p)}$ fixes some
nonzero vector $w\in W_{n}$, and hence it fixes $(w,\dots,w)\in W_{n}^{\oplus
d-1}$ as well.

  2. (ii)

Assume $k\geq 2$. We distinguish two cases.

    1. (a)

The number $n_{k}$ is not a power of $p$. Then by (i) we know that
$\mathfrak{S}_{n_{k}}^{(p)}$ fixes a nonzero point $v\in W_{n_{k}}^{\oplus
d-1}$, so the nonzero point

$(0,\dots,0;v)\in(W_{k-1}({d};{{\bf n}^{\prime}}))^{\oplus n_{k}}\oplus
W_{n_{k}}^{\oplus d-1}=W_{k}({d};{\bf n})$

is fixed by $\mathscr{S}_{k}({\bf n})^{(p)}$.

    2. (b)

The numbers $n_{1},\dots,n_{k-1}$ are not all powers of $p$. Then, by
induction hypothesis, there is a nonzero vector $V\in W_{k-1}({d};{{\bf
n}^{\prime}})$ fixed by $\mathscr{S}_{k-1}({{\bf n}^{\prime}})^{(p)}$, so the
nonzero vector

$(V,\dots,V;0)\in(W_{k-1}({d};{{\bf n}^{\prime}}))^{\oplus n_{k}}\oplus
W_{n_{k}}^{\oplus d-1}=W_{k}({d};{\bf n})$

is fixed by $\mathscr{S}_{k}({\bf n})^{(p)}$.

This completes the proof of the lemma. ∎

Now, we give the third proof of the existence part of Theorem 1.3.

######  Corollary 5.3 (Existence in Theorem 1.3).

Let $d\geq 1$, $k\geq 1$ and $n_{1},\dots,n_{k}\geq 2$ be integers. Assume
that $n_{1},\dots,n_{k}$ are not all powers of the same prime number. Then,
there exists an $\mathscr{S}_{k}({n_{1},\dots,n_{k}})$-equivariant map

$C_{k}({d};{n_{1},\dots,n_{k}})~{}\longrightarrow~{}S(W_{k}({d};{n_{1},\dots,n_{k}})).$

###### Proof.

As already noted in the beginning of the section, the existence of an
equivariant map is equivalent to vanishing of the obstruction class

$\mathfrak{o}_{\mathscr{S}_{k}({\bf n})}\in H^{M_{k}}_{\mathscr{S}_{k}({\bf
n})}(\mathscr{C}_{k}({d};{\bf n});\mathscr{Z}_{k}).$

Let $p$ be a fixed prime number dividing the order $|\mathscr{S}_{k}({\bf
n})|$. Since $n_{1},\dots,n_{k}$ are not all powers of the same prime number
by Lemma 5.2 we have

$\mathfrak{o}_{\mathscr{S}_{k}({\bf n})^{(p)}}=0\in
H^{M_{k}}_{\mathscr{S}_{k}({\bf n})^{(p)}}(\mathscr{C}_{k}({d};{\bf
n});\mathscr{Z}_{k}).$

Next, the restriction morphism

$\mathrm{res}\colon H^{M_{k}}_{\mathscr{S}_{k}({\bf
n})}(\mathscr{C}_{k}({d};{\bf n});\mathscr{Z}_{k})\longrightarrow
H^{M_{k}}_{\mathscr{S}_{k}({\bf n})^{(p)}}(\mathscr{C}_{k}({d};{\bf
n});\mathscr{Z}_{k}),$

and the transfer morphism

$\mathrm{trf}\colon H^{M_{k}}_{\mathscr{S}_{k}({\bf
n})^{(p)}}(\mathscr{C}_{k}({d};{\bf n});\mathscr{Z}_{k})\longrightarrow
H^{M_{k}}_{\mathscr{S}_{k}({\bf n})}(\mathscr{C}_{k}({d};{\bf
n});\mathscr{Z}_{k})$

satisfy

$\mathrm{trf}\circ\mathrm{res}=[\mathscr{S}_{k}({\bf n}):\mathscr{S}_{k}({\bf
n})^{(p)}]\cdot\operatorname{\mathrm{id}}$

due to [5, Lem. 5.4]. Moreover, $\mathrm{res}$ maps the obstruction class
$\mathfrak{o}_{\mathscr{S}_{k}({\bf n})}$ to the restricted obstruction class
$\mathfrak{o}_{\mathscr{S}_{k}({\bf n})^{(p)}}$, that is

$\mathrm{res}(\mathfrak{o}_{\mathscr{S}_{k}({\bf
n})})=\mathfrak{o}_{\mathscr{S}_{k}({\bf n})^{(p)}}.$

Hence, it follows that

$[\mathscr{S}_{k}({\bf n}):\mathscr{S}_{k}({\bf
n})^{(p)}]\cdot\mathfrak{o}_{\mathscr{S}_{k}({\bf
n})}=(\mathrm{trf}\circ\mathrm{res})(\mathfrak{o}_{\mathscr{S}_{k}({\bf
n})})=\mathrm{trf}(\mathfrak{o}_{\mathscr{S}_{k}({\bf n})^{(p)}})=0.$ (28)

Since we have

$\gcd\big{\\{}[\mathscr{S}_{k}({\bf n}):\mathscr{S}_{k}({\bf
n})^{(p)}]:~{}p\text{ is a prime dividing }|\mathscr{S}_{k}({\bf
n})|\big{\\}}=1,$

there exists a $\mathbb{Z}$-linear combination

$\sum_{p\big{|}|\mathscr{S}_{k}({\bf n})|\text{
prime}}x_{p}\cdot[\mathscr{S}_{k}({\bf n}):\mathscr{S}_{k}({\bf
n})^{(p)}]~{}=~{}1.$

Finally, this equality and (28) imply

$\mathfrak{o}_{\mathscr{S}_{k}({\bf n})}=\sum_{p\big{|}|\mathscr{S}_{k}({\bf
n})|\text{ prime}}x_{p}\cdot[\mathscr{S}_{k}({\bf n}):\mathscr{S}_{k}({\bf
n})^{(p)}]\cdot\mathfrak{o}_{\mathscr{S}_{k}({\bf n})}~{}=~{}0,$

as desired. ∎

##  6\. Continuity of partitions

The set $\mathscr{K}^{d}$ of convex bodies in $\mathbb{R}^{d}$ with non-empty
interior is a metric space with _Hausdorff metric_ given by

$\operatorname{\textrm{d}_{\textrm{H}}}(X,Y):=\max\\{\sup_{x\in
X}\operatorname{\textrm{dist}}(x,Y),\,\sup_{y\in
Y}\operatorname{\textrm{dist}}(y,X)\\}.$

_Symmetric difference metric_ on $\mathscr{K}^{d}$, given by

$\operatorname{\textrm{d}_{\textrm{S}}}(X,Y):=\mathcal{L}^{d}(X\triangle Y),$

where $\mathcal{L}^{d}$ is the Lebesgue measure on $\mathbb{R}^{d}$, induces
the same topology on $\mathscr{K}^{d}$ (see [14]). Hence convergence of a
sequence in one metric is equivalent to convergence in another.

The main result of the section is Theorem 6.6.

###  6.1. Existence of weights

We start with the definition of the generalised Voronoi diagram.

######  Definition 6.1.

For $x:=(x_{1},\dots,x_{n})\in F(\mathbb{R}^{d},n)$ and
$w:=(w_{1},\dots,w_{n})\in\mathbb{R}^{n}$ let us denote by

$C(x,w):=(C_{1}(x,w),\dots,C_{n}(x,w))$

the generalised Voronoi diagram with sites $x$ and weights $w$, where

$C_{i}(x,w):=\\{p\in\mathbb{R}^{d}:~{}\|p-x_{i}\|^{2}-w_{i}\leq\|p-x_{j}\|^{2}-w_{j}~{}~{}\text{
for all }1\leq j\leq n\\}$ (29)

denotes the $i$the Voronoi cell, for each $1\leq i\leq n$.

Region $C_{i}(x,w)$ can be considered as the set of points
$p\in\mathbb{R}^{d}$ where the affine function

$f_{i}(p):=\|p-x_{i}\|^{2}-\|p\|^{2}-w_{i}$

takes minimal value among all functions $f_{j}$ of the same type. In
particular, regions $C_{i}(x,w)$ have disjoint interiors and cover the whole
$\mathbb{R}^{d}$.

Notice that $C(x,w)=C(x,w+(t,\dots,t))$ for any $t\in\mathbb{R}$, so we will
usually restrict our attention to weights $w\in W_{n}\subseteq\mathbb{R}^{n}$.

######  Definition 6.2.

For a convex body $K\in\mathscr{K}^{d}$, sites $x\in F(\mathbb{R}^{d},n)$ and
weights $w\in W_{n}$, let us denote by

$K(x,w):=(K_{1}(x,w),\dots,K_{n}(x,w))$

the partition of $K$ given by $K_{i}(x,w):=K\cap C_{i}(x,w)$ for each $1\leq
i\leq n$.

As before, for $K\in\mathscr{K}^{d}$ and a probability measure $\mu$ on
$\mathbb{R}^{d}$ which is absolutely continuous with respect to the Lebesgue
measure, let
$\operatorname{\mathrm{EMP}}_{\mu}(K,n)\subseteq(\mathscr{K}^{d})^{\times n}$
denote the space of equal mass partitions of $K$ with respect to the measure
$\mu$.

######  Lemma 6.3 (Existence of weights, [12, Thm. 1]).

Let $d\geq 1$ and $n\geq 1$ be integers, $\lambda\in(0,1]^{n}$ a vector with
$\lambda_{1}+\dots+\lambda_{n}=1$. Let $K\in\mathscr{K}^{d}$ be a convex body,
and let $\mu$ be a probability measure on $\mathbb{R}^{d}$ which is absolutely
continuous with respect to the Lebesgue measure. For any choice of sites $x\in
F(\mathbb{R}^{d},n)$ there exist unique weights $w_{K;\lambda}(x)\in W_{n}$
such that the generalised Voronoi diagram $C(x,w_{K;\lambda}(x))$ produces a
partition of $K$

$K(x,w_{K;\lambda}(x))=K\cap C(x,w_{K;\lambda}(x))\in(\mathscr{K}^{d})^{\times
n}$

such that

$\mu(K_{i}(x,w_{K;\lambda}(x)))=\lambda_{i}=\lambda_{i}\cdot\mu(K)$

for each $1\leq i\leq n$.

In particular, for the unique weight vector
$w_{K}(x):=w_{K;(\frac{1}{n},\dots,\frac{1}{n})}(x)$ we have

$K(x):=K(x,w_{K}(x))\in\operatorname{\mathrm{EMP}}_{\mu}(K,n).$

The relevant case for this paper is when
$\lambda=(\frac{1}{n},\dots,\frac{1}{n})$. Then, the equal mass partition
$K(x)=(K_{1}(x),\dots,K_{n}(x))$ of a convex body $K$ with respect to $\mu$
guaranteed by the previous lemma is called the regular equipartition of $K$
with sites $x=(x_{1},\dots,x_{n})$.

In [17, Sec. 2] and [7, Sec. 2] the existence of unique weights from the lemma
is proved via the theory of optimal transport. We include a topological proof
of this fact developed by Moritz Firsching, which relies on some results from
[12].

Let us denote by $e_{1},\dots,e_{n}\in\mathbb{R}^{n}$ the standard basis
vectors, and let
$\Delta_{n-1}:=\operatorname{\textrm{conv}}\\{e_{1},\dots,e_{n}\\}$ be the
standard $(n-1)$-dimensional simplex. The faces of $\Delta_{n-1}$ are convex
sets of the form $\operatorname{\textrm{conv}}\\{e_{i}:i\in I\\}$ for
$I\subseteq[n]$. The first part of the proof of Lemma 6.3 is the following
lemma.

######  Lemma 6.4.

Let $f:\Delta_{n-1}\longrightarrow\Delta_{n-1}$ be a continuous self map of
the simplex. Then, the following statements hold.

  1. (i)

Map $f$ is surjective if $f(\sigma)\subseteq\sigma$ holds for all faces
$\sigma\subseteq\Delta_{n-1}$.

  2. (ii)

Map $f$ is injective on the preimage
$f^{-1}(\operatorname{\mathrm{int}}(\Delta_{n-1}))$ of the interior if
additionaly the following condition holds:

     * For a non-empty subset $I\subsetneq[n]$, a point $x\in\operatorname{\mathrm{int}}(\Delta_{n-1})$, $t:=\sum_{i\in I}x_{i}$, $\alpha\in(t,1]$ and

$x^{I}_{\alpha}:=\alpha\left(\sum_{i\in
I}\frac{x_{i}}{t}e_{i}\right)+(1-\alpha)\left(\sum_{j\in[n]\setminus
I}\frac{x_{j}}{1-t}e_{j}\right)\in\Delta_{n-1}.$

Then, for every $x\in\operatorname{\mathrm{int}}(\Delta_{n-1})$, every non-
empty subset $I\subsetneq[n]$, every $\alpha\in(t,1]$ and every index $i\in I$
we have $f_{i}(x^{I}_{\alpha})\geq f_{i}(x)$ such that for at least one index
$i\in I$ the inequality is strict, where $f_{i}$ denotes the $i$th coordinate
function of $f$.

###### Proof due to M. Firsching.

(i) The map $f$ is homotopic to the identity via $f_{\tau}:=\tau
f+(1-\tau)\operatorname{\mathrm{id}}_{\Delta_{n-1}}$ for $\tau\in[0,1]$. Each
map $f_{\tau}$ in the homotopy satisfies $f_{\tau}(\sigma)\subseteq\sigma$ and
hence induces a homotopy of quotient maps

$\widetilde{f}\simeq\operatorname{\mathrm{id}}_{\Delta_{n-1}/\partial\Delta_{n-1}}:\Delta_{n-1}/\partial\Delta_{n-1}\longrightarrow\Delta_{n-1}/\partial\Delta_{n-1}.$

Since $\Delta_{n-1}/\partial\Delta_{n-1}\approx S^{n-1}$, the $\widetilde{f}$
is not nullhomotopic and hence is surjective, since every non-surjective self
map of the sphere is necessarily nullhomotopic. The surjectivity of the
quotient map $\widetilde{f}$ implies the surjectivity of $f$ by the fact that
$\operatorname{\mathrm{int}}(\Delta_{n-1})=\Delta_{n-1}\setminus\partial\Delta_{n-1}$
is dense in $\Delta_{n-1}$.

(ii) The injectivity argument is similar to the one in [12, Thm. 1]. Suppose
for two different points $x,y\in\Delta_{n-1}$ we have
$f(x)=f(y)\in\operatorname{\mathrm{int}}(\Delta_{n-1})$. Then
$x,y\in\operatorname{\mathrm{int}}(\Delta_{n-1})$, since $f$ maps each face to
iteself. Define

$I(x,y):=\\{i\in[n]:~{}x_{i}/y_{i}=\min_{1\leq j\leq n}x_{j}/y_{j}\\}.$

Since $x\neq y$ we have $\emptyset\neq I\subsetneq[n]$. We will inductively
define a sequence of points

$x=x^{0},x^{1},\dots,x^{k}=y\in\Delta_{n-1}$

for some integer $k\geq 1$, which satisfy

$I({x^{0},y})\subsetneq I({x^{1},y})\subsetneq\dots\subsetneq
I({x^{k-1},y})\subsetneq[n]$

and such that for each $1\leq l\leq k$ we have $f_{i}(x^{l-1})\leq
f_{i}(x^{l})$ for all $i\in I({x^{l-1},y})$, with the strict inequality
holding true for at least one index in $i\in I({x^{l-1},y})$. Indeed, if we
prove this, than for the index $i\in I(x^{0},y)$ for which the strict
inequality $f_{i}(x^{0})<f_{i}(x^{1})$ is achieved, we would have

$f_{i}(x)=f_{i}(x^{0})<f_{i}(x^{1})\leq\dots\leq f_{i}(x^{k})=f_{i}(y),$

which contradicts $f(x)=f(y)$.

Suppose $x^{0},\dots,x^{l-1}\neq y$ with the above properties have already
been constructed, for some $l\geq 1$, and let us construct $x^{l}$. Since
$I({x^{l-1},y})\subsetneq[n]$, by the assumption of part (ii), there exists
some $\alpha$ with

$\sum_{i\in I({x^{l-1},y})}x^{l-1}_{i}<\alpha\leq 1$

such that the point $x^{l}:=x_{\alpha}^{I({x^{l-1},y})}$ satisfies
$I({x^{l-1},y})\subsetneq I({x^{l},y})$ and $f_{i}(x^{l})\leq f_{i}(x^{l})$
for all $i\in I({x^{l-1},y})$, with the strict inequality for at least one
index $i\in I({x^{l-1},y})$. If $I({x^{l},y})=[n]$, then $k:=l$ and $x^{l}=y$,
so we stop induction. Otherwise, we continue for finite more steps until
$I({x^{k},y})=[n]$.

∎

###### Proof of Lemma 6.3 due to M. Firsching.

Let us fix a point $x\in F(\mathbb{R}^{d},n)$. We want to show existence of a
weight vector $w_{K;\lambda}(x)\in W_{n}$.

Given a point $z=(z_{1},\dots,z_{n})\in\Delta_{n-1}$ set
$w(z):=(\log(z_{1}),...,\log(z_{n}))$. Here we allow $\log(0):=-\infty$ and
extend the definition of $C(x,w)$ to include weight vectors $w$ with some (but
not all) coordinates being $-\infty$. We define the continuous map $m$ by

$m\colon\Delta_{n-1}\longrightarrow\Delta_{n-1},\qquad z\longmapsto
m(z):=(\mu(K_{1}(x,w(z))),\dots,\mu(K_{n}(x,w(z)))).$

If this map is surjective and injective on the preimage
$m^{-1}(\operatorname{\mathrm{int}}(\Delta_{n-1}))$, we can set
$w_{K;\lambda}(x):=w(z)$, where $\\{z\\}=m^{-1}(\\{\lambda\\})$ is the unique
point in the fiber of $\lambda\in\operatorname{\mathrm{int}}(\Delta_{n-1})$.
To prove the property of the map $m$ we use Lemma 6.4.

Before doing that, let us see why the map $m$ is continuous. First, notice
that if we show it is continuous on the interior
$\operatorname{\mathrm{int}}(\Delta_{n-1})$, the continuity of $m$ on the
whole domain follows by density of the interior and the fact that the value of
$m$ on the boundary is the limit value of interior points. Thus, let $z^{k}\to
z$ as $k\to\infty$ be a convergence in
$\operatorname{\mathrm{int}}(\Delta_{n-1})$. Then by part (ii) of Lemma 6.7
for each $1\leq i\leq n$ we have

${\mathscr{L}}^{d}(K_{i}(x,w(z^{k}))\triangle
K_{i}(x,w(z)))\xrightarrow{k\to\infty}0.$

By (30) it further follows that for each $1\leq i\leq n$ we have

$\mu(K_{i}(x,w(z^{k})))\xrightarrow{k\to\infty}\mu(K_{i}(x,w(z))),$

proving ultimately that $m(z^{k})\to m(z)$ as $k\to\infty$.

To prove surjectivity of $m$, by Lemma 6.4 part (i) it is enough to show that
$m$ maps each face to itself. Indeed, assume for $z\in\Delta_{n-1}$ we have
$z_{i}=0$ for some $1\leq i\leq n$. Then the $i$th coordinate of $w(z)$ is
$-\infty$ and the $i$th coordinate of $m(z)$ is zero, since the $i$th Voronoi
region $C_{i}(x,w(z))$ is empty. Therefore a face of $\Delta_{n-1}$ is mapped
to itself by $m$.

For injectivity of $m$ on the preimage
$m^{-1}(\operatorname{\mathrm{int}}(\Delta_{n-1}))$ we use Lemma 6.4 part
(ii). We want to show that for any point $z\in\Delta_{n-1}$, nonempty
$I\subsetneq[n]$, $t:=\sum_{i\in I}z_{i}$ and $\alpha\in(t,1]$ the condition
in part (ii) is satisfied. Notice that the coordinate function of $w$
satisfies

$w_{i}(z_{\alpha}^{I})=\log(\alpha/t)+w_{i}(z),~{}~{}\text{ for }i\in I$

for $i\in I$ and

$w_{j}(z_{\alpha}^{I})=\log((1-\alpha)/(1-t))+w_{j}(z),~{}~{}\text{ for
}j\in[n]\setminus I.$

If we denote by $w^{\prime}$ the weight vector obtained from
$w(z_{\alpha}^{I})$ by adding a constant

$-\log((1-\alpha)/(1-t))=\log((1-t)/(1-\alpha))$

to each coordinate, we have equality of Voronoi diagrams

$C(x,w(z_{\alpha}^{I}))=C(x,w^{\prime}),$

so one can use the weight vector $w^{\prime}$ to calculate
$m(z_{\alpha}^{I})$. Moreover, we have

$w^{\prime}_{i}=\log(\alpha/t)+w_{i}(z)+\log((1-t)/(1-\alpha))=\log(1/t-1)-\log(1/\alpha-1)+w_{i}(z)>w_{i}(z)$

for $i\in I$ and $w^{\prime}_{j}=w_{j}(z)$ for $j\in[n]\setminus I$. In
particular, for each $i\in I$ we have

$C_{i}(x,w(z))\subseteq C_{i}(x,w^{\prime})=C_{i}(x,w(z_{\alpha}^{I})),$

which implies $m_{i}(z_{\alpha}^{I})\geq m_{i}(z)$. Therefore, by [12, Lem. 2]
it follows that there exists an index $i\in I$ such that
$m_{i}(z_{\alpha}^{I})>m_{i}(z)$. Hence, the assumption in part (ii) of Lemma
6.4 holds. ∎

###  6.2. Continuity of partitions

Lemma 6.3 implies that, for a given $K\in\mathscr{K}^{d}$, the function

$F(\mathbb{R}^{d},n)\longrightarrow W_{n},~{}~{}~{}x\longmapsto w_{K}(x)$

exists and is $\mathfrak{S}_{n}$-equivariant. In [15, Lem. 3] it was shown
that the function is continuous on parameter $x\in F(\mathbb{R}^{d},n)$. Since
the function also depends on the convex body $K\in\mathscr{K}^{d}$ it is
natural to ask: Is the fuctnion $w_{K}(x)$ continuous in parameters $(x,K)\in
F(\mathbb{R}^{d},n)\times\mathscr{K}^{d}$?

We give the positive answer to this question in the lemma which follows. For
the proof see Section 6.3.

######  Lemma 6.5 (Continuity of weights).

Let $\mu$ be a probability measure on $\mathbb{R}^{d}$ which is absolutely
continuous with respect to the Lebesgue measure. The assignment

$F(\mathbb{R}^{d},n)\times\mathscr{K}^{d}\longrightarrow
W_{n},~{}(x,K)\longmapsto w_{K}(x),$

given by Lemma 6.3, is continuous and $\mathfrak{S}_{n}$-equivariant, where
the $\mathfrak{S}_{n}$-action on the $\mathscr{K}^{d}$-coordinate in the
domain is set to be trivial.

For a given convex body $K\in\mathscr{K}^{d}$, the continuity of the
assignment

$F(\mathbb{R}^{d},n)\longrightarrow\operatorname{\mathrm{EMP}}_{\mu}(K,n),\qquad
x\longmapsto K(x)=K(x,w_{K}(x)),$

given by Lemma 6.3, is proved in [17, Thm. 2.1] and [7, Thm. 2.1]. Again as
before, this assignment depends on $K$, so the next natural question arises:
Is $K(x)$ continuous in parameters $(x,K)\in
F(\mathbb{R}^{d},n)\times\mathscr{K}^{d}$?

The main result of this section is the following theorem which gives the
positive answer to the previous question.

######  Theorem 6.6 (Continuity of partitions).

The function

$F(\mathbb{R}^{d},n)\times\mathscr{K}^{d}\longrightarrow(\mathscr{K}^{d})^{\times
n},\qquad(x,K)\longmapsto K(x),$

where $K(x)$ is the regular partition of $K$ with sites $x$ from Lemma 6.3, is
a continuous $\mathfrak{S}_{n}$-equivariant map which satisfies the
restriction property

$F(\mathbb{R}^{d},n)\times\\{K\\}\longrightarrow\operatorname{\mathrm{EMP}}_{\mu}(K,n),\qquad(x,K)\longmapsto
K(x),$

for each $K\in\mathscr{K}^{d}$.

###### Proof.

To show the claim, it is enough to show that for each $1\leq i\leq n$ the
coordinate map

$F(\mathbb{R}^{d},n)\times\mathscr{K}^{d}\longrightarrow\mathscr{K}^{d},\qquad(x,K)\longmapsto
K_{i}(x,w_{K}(x))=K\cap C_{i}(x,w_{K}(x))$

is continuous, where weights $w_{K}(x)$ are given by Lemma 6.3, and region
$C_{i}(x,w_{K}(x))$ is the $i$th component (29) of the generalised Voronoi
diagram $C(x,w_{K}(x))$ with sites $x$ and weights $w_{K}(x)$. To show
sequential continuity, let

$(x^{k},K^{k})\xrightarrow{~{}~{}k\to\infty~{}~{}}(x,K)\in
F(\mathbb{R}^{d},n)\times\mathscr{K}^{d}$

be a converging sequence, and let us denote the weights by $w:=w_{K}(x)$ and
$w^{k}:=w_{K^{k}}(x^{k})$. By Lemma 6.5, we have $w^{k}\to w$ as $k\to\infty$.

Moreover, since $K_{i}(x,w)=K\cap C_{i}(x,w)$ has a non-empty interior and
$(x^{k},w^{k})\to(x,w)$ as $k\to\infty$, it follows that

$K_{i}(x^{k},w^{k})=K\cap C_{i}(x^{k},w^{k})$

has a non-empty interior as well, hence
$K_{i}(x^{k},w^{k})\in\mathscr{K}^{d}$. Therefore, all notions used in the
following string of inequalities are well defined. We have

$\displaystyle\operatorname{\textrm{d}_{\textrm{S}}}(K_{i}(x,w),K^{k}_{i}(x^{k},w^{k}))$
$\displaystyle\leq\operatorname{\textrm{d}_{\textrm{S}}}(K_{i}(x,w),K_{i}(x^{k},w^{k}))+\operatorname{\textrm{d}_{\textrm{S}}}(K_{i}(x^{k},w^{k}),K^{k}_{i}(x^{k},w^{k}))$
$\displaystyle\leq\operatorname{\textrm{d}_{\textrm{S}}}(K_{i}(x,w),K_{i}(x^{k},w^{k}))+\operatorname{\textrm{d}_{\textrm{S}}}(K,K^{k})\xrightarrow{~{}~{}k\to\infty~{}~{}}0,$

because the first summand in the last row tends to zero as $k\to\infty$ by
Lemma 6.7 (ii), so the continuity follows. ∎

###  6.3. Auxiliar lemmas

For a vector $(a,b)\in\mathbb{R}^{d}\times\mathbb{R}$ such that $a\neq
0\in\mathbb{R}^{d}$ let us denote by

$H_{(a,b)}:=\\{p\in\mathbb{R}^{d}\ :\ \langle a,p\rangle+b=0\\}$

the affine hyperplane induced by $(a,b)$, and by

$H^{+}_{(a,b)}:=\\{p\in\mathbb{R}^{d}\ :\ \langle a,p\rangle+b\geq 0\\}$

the corresponding closed half-space.

######  Lemma 6.7.

Let $K\in\mathscr{K}^{d}$. Then, the following statements are true.

  1. (i)

Assume the convergence

$(a^{k},b^{k})\xrightarrow{~{}k\to\infty~{}}(a,b)\in\mathbb{R}^{d}\times\mathbb{R}$

with $a,a^{k}\neq 0$, as well as

$K\cap H_{(a,b)}^{+},~{}K\cap H_{(a^{k},b^{k})}^{+}\in\mathscr{K}^{d}$

for each $k\geq 1$. Then

$K\cap
H_{(a^{k},b^{k})}^{+}\xrightarrow{~{}\operatorname{\textrm{d}_{\textrm{S}}}~{}}K\cap
H_{(a,b)}^{+}$

as $k\to\infty$.

  2. (ii)

Assume the convergence

$(x^{k},w^{k})\xrightarrow{~{}k\to\infty~{}}(x,w)\in F(\mathbb{R}^{d},n)\times
W_{n},$

and let $1\leq i\leq n$, as well as

$K_{i}(x,w),K_{i}(x^{k},w^{k})\in\mathscr{K}^{d}$

for each $k\geq 1$. Then

$K_{i}(x^{k},w^{k})\xrightarrow{~{}\operatorname{\textrm{d}_{\textrm{S}}}~{}}K_{i}(x,w)$

as $k\to\infty$.

###### Proof.

(i) Since $\operatorname{\textrm{d}_{\textrm{S}}}$ and
$\operatorname{\textrm{d}_{\textrm{H}}}$ induce the same topology on
$\mathscr{K}$, it is enough to prove convergence in
$\operatorname{\textrm{d}_{\textrm{H}}}$ metric. Recall that

$\operatorname{\textrm{d}_{\textrm{H}}}(K\cap H_{(a^{k},b^{k})}^{+},K\cap
H_{(a,b)}^{+})=\max\big{\\{}\sup_{x\in K\cap
H_{(a^{k},b^{k})}^{+}}\operatorname{\textrm{dist}}(x,K\cap
H_{(a,b)}^{+}),\sup_{x\in K\cap
H_{(a,b)}^{+}}\operatorname{\textrm{dist}}(x,K\cap
H_{(a^{k},b^{k})}^{+})\big{\\}}.$

For the first supremum we have

$\sup_{x\in K\cap H_{(a^{k},b^{k})}^{+}}\operatorname{\textrm{dist}}(x,K\cap
H_{(a,b)}^{+})\xrightarrow{~{}k\to\infty~{}}0.$

Indeed, assume to the contrary that for some sequence $(z_{k}\in K\cap
H_{(a^{k},b^{k})}^{+})_{k\geq 1}$, we have that

$\operatorname{\textrm{dist}}(z_{k},K\cap H_{(a,b)}^{+})>2\varepsilon.$

Since $K$ is compact, we could have chosen the sequence $z_{k}$ from the very
start such that $z_{k}\xrightarrow{~{}k\to\infty~{}}z\in K$, and so

$\operatorname{\textrm{dist}}(z,K\cap H_{(a,b)}^{+})>\varepsilon.$

From this we obtain a contradiction by showing $z_{k}\notin
H_{(a^{k},b^{k})}^{+}$ for $k>>0$. This is indeed true, since

$\langle z_{k},a^{k}\rangle+b^{k}\xrightarrow{~{}k\to\infty~{}}\langle
z,a\rangle+b<0.$

For the second supremum we have that

$\sup_{x\in K\cap H_{(a,b)}^{+}}\operatorname{\textrm{dist}}(x,K\cap
H_{(a^{k},b^{k})}^{+})\xrightarrow{~{}k\to\infty~{}}0.$

Indeed, assume to the contrary that for some sequence $(z_{k}\in K\cap
H_{(a,b)}^{+})_{k\geq 1}$ we have

$\operatorname{\textrm{dist}}(z_{k},K\cap
H_{(a^{k},b^{k})}^{+})>2\varepsilon.$

Since $K$ is compact, there is a subsequence with
$z_{k}\xrightarrow{~{}k\to\infty~{}}z\in K\cap H_{(a,b)}^{+}$, hence

$\operatorname{\textrm{dist}}(z,K\cap H_{(a^{k},b^{k})}^{+})>\varepsilon.$

for each $k\geq 1$. Notice first that $z\in K\cap H_{(a,b)}$. Indeed, strict
inequality $z\cdot a+b>0$ would

$\langle z,a^{k}\rangle+b^{k}\xrightarrow{~{}k\to\infty~{}}\langle
z,a\rangle+b>0,$

which would mean $z\in K\cap H_{(a^{k},b^{k})}^{+}$, which is impossible.
Since $(H^{+}_{(a,b)}\setminus H_{(a,b)})\cap K\neq\emptyset$ by assumption,
by convexity there exists

$y\in(H^{+}_{(a,b)}\setminus H_{(a,b)})\cap K$

with $\operatorname{\textrm{dist}}(x,y)<\varepsilon$. Moreover, we have $y\in
K\cap H_{(a^{k},b^{k})}^{+}$ for $k>>0$, since

$\langle y,a^{k}\rangle+b^{k}\xrightarrow{~{}k\to\infty~{}}\langle
y,a\rangle+b>0.$

This means that

$\varepsilon>\operatorname{\textrm{dist}}(x,y)\geq\operatorname{\textrm{dist}}(z,K\cap
H_{(a^{k},b^{k})}^{+})>\varepsilon$

for $k>>0$, which is a contradiction.

(ii) We will first show that for two converging sequences
$A^{k}\xrightarrow{~{}\operatorname{\textrm{d}_{\textrm{S}}}~{}}A$ and
$B^{k}\xrightarrow{~{}\operatorname{\textrm{d}_{\textrm{S}}}~{}}B$ in
$\mathscr{K}^{d}$ such that $A^{k}\cap B^{k},A\cap B\in\mathscr{K}^{d}$, we
have

$A^{k}\cap
B^{k}\xrightarrow{~{}\operatorname{\textrm{d}_{\textrm{S}}}~{}}A\cap B$

as $k\to\infty$. Indeed, from

$(A^{k}\cap B^{k})\triangle(A\cap B)\subseteq(A^{k}\triangle
A)\cup(B^{k}\triangle B)$

it follows that

$\operatorname{\textrm{d}_{\textrm{S}}}(A^{k}\cap B^{k},A\cap
B)\leq\operatorname{\textrm{d}_{\textrm{S}}}(A^{k},A)+\operatorname{\textrm{d}_{\textrm{S}}}(B^{k},B)\xrightarrow{~{}k\to\infty~{}}0,$

as desired.

Back to the proof of the main claim. For $(x,w)\in F(\mathbb{R}^{d},n)\times
W_{n}$ and $1\leq i\leq n$ the $i$th component $C_{i}(x,w)$ of the generalised
Voronoi diagram $C(x,w)$ is equal to the intersection of closed half-spaces

$H^{+}_{i,j}(x,w):=\\{p\in\mathbb{R}^{d}:\|p-x_{j}\|^{2}-\|x_{j}\|^{2}-w_{j}-\|p-x_{i}\|^{2}-\|x_{i}\|^{2}-w_{i}\geq
0\\}$

for $1\leq j\leq i$ and $j\neq i$. Therefore, by the repeated use of the above
intersection argument and part (i), we have

$\operatorname{\textrm{d}_{\textrm{S}}}(K_{i}(x,w),K_{i}(x^{k},w^{k}))=\operatorname{\textrm{d}_{\textrm{S}}}(K\cap\bigcap_{j\neq
i}H^{+}_{i,j}(x,w),K\cap\bigcap_{j\neq
i}H^{+}_{i,j}(x^{k},w^{k}))\xrightarrow{~{}k\to\infty~{}}0,$

which completes the proof. ∎

Recall, for a measure $\mu$ which is absolutely continuous with respect to the
Lebesgue measure ${\mathscr{L}}^{d}$ on $\mathbb{R}^{d}$, by the Radon-Nikodym
theorem, there exists an integrable function $f:\mathbb{R}^{d}\to\mathbb{R}$
such that

$\mu(A)=\int_{A}f~{}d{\mathscr{L}}^{d},$

for each measurable set $A\subseteq\mathbb{R}^{d}$. We will need the next
measure-theoretic claim.

######  Lemma 6.8.

Let $\mu$ be a probability measure on $\mathbb{R}^{d}$ which is absolutely
continuous with respect to the Lebesgue measure ${\mathscr{L}}^{d}$. Then, the
following implication holds

${\mathscr{L}}^{d}(A^{k})\xrightarrow{~{}k\to\infty~{}}0~{}~{}\implies\mu(A^{k})\xrightarrow{~{}k\to\infty~{}}0,$
(30)

where $(A^{k})_{k\geq 1}$ is any sequence of measurable sets in
$\mathbb{R}^{d}$.

###### Proof.

The proof, in essence, follows from a use of the Dominant Convergence Theorem
[1, Sec. A3.2]. Let $\chi_{A}$ denotes the characteristic function of a
measurable set $A\subseteq\mathbb{R}^{d}$. We prove (30) in two steps.

Let is first show that a sequence of functions $(f\cdot\chi_{A_{k}})_{k\geq
1}$ converges pointwise almost everywhere to the zero function. Indeed, let
$g$ denote the pointwise limit of the sequence $(f\cdot\chi_{A_{k}})_{k\geq
1}$. Let $\varepsilon>0$ and let us restrict to a subsequence such that
$\sum_{k=1}^{\infty}{\mathscr{L}}^{d}(A^{k})<\varepsilon$. Then, we have

$\\{z\in\mathbb{R}^{d}:~{}g(z)\neq 0\\}\subseteq\bigcup_{k=1}^{\infty}A^{k},$

and therefore

${\mathscr{L}}^{d}(\\{z\in\mathbb{R}^{d}:~{}g(z)\neq
0\\})\leq\sum_{k=1}^{\infty}{\mathscr{L}}^{d}(A^{k})<\varepsilon.$

Since this holds for any $\varepsilon>0$, it follows that
${\mathscr{L}}^{d}(\\{z\in\mathbb{R}^{d}:~{}g(z)\neq 0\\})=0$, so $g$ is zero
almost everywhere.

Finally, by the Dominant Convergence Theorem applied to the dominant $f$, we
have

$\lim_{k\to\infty}\mu(A^{k})=\lim_{k\to\infty}\int_{\mathbb{R}^{d}}f\cdot\chi_{A_{k}}(x)~{}d{\mathscr{L}}^{d}(x)=\int_{\mathbb{R}^{d}}\big{(}\lim_{k\to\infty}f\cdot\chi_{A_{k}}(x)\big{)}~{}d{\mathscr{L}}^{d}(x)=0,$

which finishes the proof of the implication (30). ∎

We are now in the position to prove Lemma 6.5.

###### Proof of Lemma 6.5.

Let $x^{k}\longrightarrow x\in F(\mathbb{R}^{d},n)$ and
$K^{k}\xrightarrow{~{}\operatorname{\textrm{d}_{\textrm{H}}}~{}}K\in\mathscr{K}^{d}$
as $k\to\infty$. We want to show

$w_{K^{k}}(x^{k})\xrightarrow{~{}k\to\infty~{}}w_{K}(x)\in W_{n}.$

Let us shorten the notation and denote $w:=w_{K}(x)$ and
$w^{k}:=w_{K^{k}}(x^{k})$ for each $k\geq 1$.

First, notice that the sequence $(w^{k})_{k\geq 1}$ is bounded. Indeed,
$K^{k}\xrightarrow{\operatorname{\textrm{d}_{\textrm{H}}}}K$ as $k\to\infty$
implies

$\sup_{z\in
K^{k}}\operatorname{\textrm{dist}}(z,K)\xrightarrow{~{}k\to\infty~{}}0,$

so bodies $K^{k}$ are contained in a bounded region around
$K\subseteq\mathbb{R}^{d}$. Similarly, $x^{k}\to x$ as $k\to\infty$ implies
that the sites $x^{k}$ live in a bounded region around $x\in
F(\mathbb{R}^{d},n)$. Therefore, for any $1\leq i<j\leq n$ the difference
$|w_{i}^{k}-w_{j}^{k}|$ must be bounded for all $k\geq 1$, proving that the
sequence $(w^{k})_{k\geq 1}$ is bounded in $W_{n}$. Indeed, if for some $1\leq
i<j\leq n$ we have $|w_{i}^{k}-w_{j}^{k}|\to\infty$ as $k\to\infty$, it would
imply

$K^{k}_{i}(x^{k},w^{k})=K^{k}\cap C_{i}(x^{k},w^{k})=\emptyset$

for $k>>0$, which is a contradiction.

Convergence $w^{k}\to w$ as $k\to\infty$ is equivalent to the same convergence
for each subsequence of $(w^{k})_{k\geq 1}$. Let us assume the latter is not
true and seek a contradiction. By the boundedness of weights, and after
possibly restricting to a subsequence, we have $w^{k}\longrightarrow
w^{\prime}\in W_{n}$ as $k\to\infty$, for some $w^{\prime}\neq w$. Due to the
uniqueness of the weight vector in $W_{n}$ for given sites $x$ and a convex
body $K$, contradiction would follow from the fact that

$\mu(K_{i}(x,w^{\prime}))=\mu(K_{i}(x,w))$

for each $1\leq i\leq n$.

Since ${\mathscr{L}}^{d}(K\triangle K^{k})\to 0$ as $k\to\infty$, by (30) we
have

$|\mu(K_{i}(x^{k},w^{k}))-\mu(K^{k}_{i}(x^{k},w^{k}))|\leq\mu(K_{i}(x^{k},w^{k}))\triangle
K^{k}_{i}(x^{k},w^{k}))\leq\mu(K\triangle
K^{k})\xrightarrow{~{}k\to\infty~{}}0.$

From $(x^{k},w^{k})\longrightarrow(x,w^{\prime})$ as $k\to\infty$ and Lemma
6.7 (ii) it follows that
$K_{i}(x^{k},w^{k})\xrightarrow{\operatorname{\textrm{d}_{\textrm{S}}}}K_{i}(x,w^{\prime})$
as $k\to\infty$, therefore by (30) we have

$|\mu(K_{i}(x,w^{\prime}))-\mu(K_{i}(x^{k},w^{k}))|\xrightarrow{~{}k\to\infty~{}}0.$

Putting these two convergences together we get

$|\mu(K_{i}(x,w^{\prime}))-\mu(K^{k}_{i}(x^{k},w^{k}))|\xrightarrow{~{}k\to\infty~{}}0.$
(31)

In particular, from $|\mu(K^{k})-\mu(K)|\leq\mu(K\triangle K^{k})\to 0$ as
$k\to\infty$, we get

$\mu(K^{k}_{i}(x^{k},w^{k}))=\frac{1}{n}\mu(K^{k})\xrightarrow{~{}k\to\infty~{}}\frac{1}{n}\mu(K)=\mu(K_{i}(x,w)),$

which together with (31) implies $\mu(K_{i}(x,w^{\prime}))=\mu(K_{i}(x,w))$ as
desired. ∎

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