# Gromov-Hausdorff Convergence of Smocked Measure Spaces

Hollis Williams

###### Abstract

Smocked spaces are a special class of metric spaces which were introduced as a
generalisation of pulled thread spaces. We investigate convergence of these
spaces, showing that the smocked space obtained from the Hausdorff limit of a
sequence of smocking sets is equivalent to the Gromov-Hausdorff limit of the
corresponding smocked spaces and also obtaining an analogue of Gromov’s
compactness theorem for smocked spaces. We then introduce smocked metric
measure spaces and smocked Wasserstein spaces of measures, finding that there
is a result for Gromov-Hausdorff convergence of certain smocked Wasserstein
spaces which may be useful for studying a large class of PDEs which appear in
mathematical physics.

##  1 Introduction

The concept of a pulled thread space has existed for some time in metric
geometry [1]. Roughly speaking, the idea is to take Euclidean space
$\mathbb{E}^{N}$, view the space as a piece of cloth, sew along an interval or
some other compact connected set in that space, and then pull the thread tight
to obtain a new metric space where the sewing interval is now viewed as a
point (we will not give the formal definition as it follows as a special case
from Definition 1.1). Building on this concept, Sormani et al. recently
introduced the notion of a smocked metric space and studied the balls and
geodesics for several such spaces [2]. The definition is as follows:

Definition 1.1 Start with a finite or countable collection of disjoint
connected compact sets in $\mathbb{E}^{N}$ called smocking stitches

$I=\\{I_{j}:j\in J\\}.$

Also, assume the existence of a positive smocking separation factor $\delta$

$\delta=\text{min}\\{|z-z^{\prime}|:z\in I_{j},z^{\prime}\in
I_{j^{\prime}},j\neq j^{\prime}\in J\\}>0.$

Using these one can define a smocked metric space $(X,d)$ where each stitch in
the collection is viewed as a single point:

$X=\\{x:x\in\mathbb{E}^{n}\>\backslash\>S\\}\cup I,$

where the smocking set $S$ is

$S=\bigcup_{j\in J}I_{j}.$

We also have a smocking map $\pi:\mathbb{E}^{N}\rightarrow X$ defined by

$\pi(x)=\left\\{\begin{array}[]{ll}x&\mbox{if
}x\in\mathbb{E}^{n}\>\backslash\>S\\\ I_{j}&\mbox{if }x\in I_{j},\>j\in J\\\
\end{array}\right.$

for some index set $J$ and a smocked distance function $d:X\times
X\rightarrow[0,\infty)$ which is defined for $x,y\notin\pi(S)$ and stitches
$I_{m}$ and $I_{k}$

$d(x,y)=\text{min}\\{d_{0}(x,y),d_{1}(x,y),...\\},$

$d(x,I_{k})=\text{min}\\{d_{0}(x,z),d_{1}(x,z),...:z\in I_{k}\\},$

$d(I_{m},I_{k})=\text{min}\\{d_{0}(z^{\prime},z),d_{1}(z^{\prime},z),...:z^{\prime}\in
I_{m},z\in I_{k}\\},$

where $d_{k}$ is the sum of lengths of segments which go between $k$ stitches:

$d_{k}(v,w)=\text{min}\\{|v-z_{1}|+\sum_{i=1}^{k-1}|z^{\prime}_{i}-z_{i+1}|+|z^{\prime}_{k}-w|:z_{i},z^{\prime}_{i}\in
I_{j_{i}},j_{1}\neq...\neq j_{k}\in J\\},$

for any $v,w\in\mathbb{E}^{n}$. The smocking pseudometric
$\overline{d}:\mathbb{E}^{N}\times\mathbb{E}^{N}\rightarrow[0,\infty)$ is
defined as

$\overline{d}(v,w)=d(\pi(v),\pi(w))=\text{min}\\{d_{k}(v^{\prime},w^{\prime}):\pi(v)=\pi(v^{\prime}),\pi(w)=\pi(w^{\prime}),k\in\mathbb{N}\\}.$

One can prove that these spaces are well-defined as metric spaces and meet all
the relevant axioms [2].

There remain a number of basic questions about convergence of these metric
spaces and their subsequences, which we will now attempt to address. One of
the immediate applications mentioned in [2] is to apply results on Sormani-
Wenger intrinsic flat convergence of smocked spaces to an open question of
Gromov and Sormani on tori of almost nonnegative scalar curvature [3, 4]. In
this article, we will not consider intrinsic flat convergence and will instead
prove some rigorous results for other types of convergence (including
classical Gromov-Hausdorff convergence). It would certainly be desirable to
have some more results of this kind before embarking on further studies of the
rich geometry of smocked spaces and their tangent cones [5].

##  2 Relation between Hausdorff and Gromov-Hausdorff Convergence

We start with a result which relates Hausdorff convergence of smocking sets to
Gromov-Hausdorff convergence of the corresponding smocked spaces.

Proposition 2.1: If one defines a sequence of smocked spaces $X_{k}$ using
smocking sets $S_{k}\subset\mathbb{E}^{N}$ such that the sets Hausdorff
converge in the limit to a smocking set $S_{\infty}$, then the corresponding
smocked spaces converge to $X_{\infty}$ in the Gromov-Hausdorff sense.

Proof: Smocked spaces are complete and noncompact (see Thm 6.1 of [2]), so we
require the notion of pointed Gromov-Hausdorff convergence as defined by
Gromov.

Definition 2.2 If one has a sequence of complete noncompact metric spaces
$(X_{j},d_{j})$ and points $x_{j}\in X_{j}$, pointed Gromov-Hausdorff
convergence is defined as follows:

$(X_{j},x_{j},d_{j})\xrightarrow{pGH}(X_{\infty},x_{\infty},d_{\infty})$

if and only if for every $R>0$ the closures of the balls of radius $R$ in
$X_{j}$ converge in the Gromov-Hausdorff sense as metric spaces with the
restriction of the distance function to closures of balls in $X_{\infty}$

$(\overline{B}_{R}(x_{j}),d_{j})\xrightarrow{GH}(\overline{B}_{R}(x_{\infty}),d_{\infty}).$

Definition 2.3: Given two metric spaces $X$ and $Y$, the Gromov-Hausdorff
distance is defined to be

$d_{GH}(X,Y)=\text{inf}\bigg{\\{}d_{H}^{Z}(f(X),f(Y))\bigg{\\}}$

taken over all metric spaces $Z$ and isometric embeddings $f:X\rightarrow Z$
and $g:Y\rightarrow Z$. $d_{H}^{Z}$ is the Hausdorff distance between subsets
of $Z$

$d_{H}^{Z}(A,B)=\text{inf}\\{\epsilon>0:B\subset
T_{\epsilon}(A)\>\text{and}\>A\subset T_{\epsilon}(B)\\},$

where $T_{\epsilon}(A)=\\{x\in Z:d_{Z}(x,A)<\epsilon\\}$.

As an explicit example, start with $\mathbb{E}^{N}$ and take the sequence of
smocking sets $S_{k}$ to be the sequence of closed balls about the origin with
radius $1/k$. In the limit, $S_{\infty}$ is the point at the origin, so
$X_{\infty}=\mathbb{E}^{N}$, since we have removed a point and replaced it
with a point. Now for the sequence of smocked spaces, consider $X_{k}$:

$X_{k}=\\{x\in\mathbb{E}^{N}\backslash\overline{B}_{1/k}(0)\\}\cup\\{\overline{B}_{1/k}(0)\\}.$

In this space $\overline{B}_{1/k}(0)$ is identified with a point, so the
distance between two points $x$ and $y$ is the minimum of the direct distance
between $x$ and $y$ and all the possible distances between $x$ and
$z\in\overline{B}_{1/k}(0)$ plus the distances between
$z^{\prime}\in\overline{B}_{1/k}(0)$ and $y$. We are trying to show that the
Gromov-Hausdorff limit is $\mathbb{E}^{N}$, which amounts to showing that

$\lim_{k\rightarrow\infty}d_{GH}\big{(}\big{(}\overline{B}_{r}(x_{k}),d_{X_{k}}\big{)},\big{(}\overline{B}_{r}(0),d_{E}\big{)}\big{)}=0,$

where $d_{E}$ is the Euclidean distance function. Actually, the notion of
pointed Gromov-Hausdorff convergence is not necessary here because of Lemma
7.7 of [2], which states that for an $N$-dimensional pulled thread space $X$
where the thread is a ball of diameter $D$

$d_{GH}((X,d_{X}),(\mathbb{E}^{N},d_{E}))\leq 2D.$

(The lemma is proved by finding a suitable correspondence). However, in the
limit the diameter of the ball goes to zero, so we have

$\lim_{k\rightarrow\infty}d_{GH}((X_{k},d_{X_{k}}),(\mathbb{E}^{N},d_{E}))=0$

as required.

In general, the Gromov-Hausdorff limit $X_{\infty}$ will not necessarily be
$\mathbb{E}^{N}$ and there will be a possibly countable collection of disjoint
compact connected sets in each smocking set in the sequence, so the question
is whether an argument with correspondences still goes through in the general
case. We now need to show that for all $r>0$

$\lim_{k\rightarrow\infty}d_{GH}\big{(}\big{(}B_{r}(x_{0}^{(k)}),d_{X_{k}}\big{)},\big{(}B_{r}(x_{0}^{(\infty)}),d_{X_{\infty}}\big{)}\big{)}=0,$

where $X_{\infty}$ is the smocked space obtained from the Hausdorff limit
$S_{\infty}$ and $x_{0}^{(k)}$ is a base point in $X_{k}$. For convenience, we
will recall the definition of a correspondence.

Definition 2.4 Given two metric spaces $X$ and $Y$, a correspondence $C$ is a
set $C\subset X\times Y$ such that

$\forall x\in X\>\exists y\in Y\>\text{such that}\>(x,y)\in
C\>\text{and}\>\forall y\in Y\>\exists x\in X\>\text{such that}\>(x,y)\in C.$

To deal with the general case, consider the smocking maps
$\pi_{k}:\mathbb{E}^{N}\rightarrow X_{k}$ and
$\pi_{\infty}:\mathbb{E}^{N}\rightarrow X_{\infty}$ associated to $X_{k}$ and
$X_{\infty}$, respectively. By shifting the smocking set $S_{k}$, we can
assume that

$\pi_{k}(0)=x_{0}^{(k)}\>\>\>\>\>\>\text{for any}\>\pi_{k},$

with a similar statement holding for $S_{\infty}$. We then define a subset
$U_{r}(x_{0}^{(k)})$ of $\mathbb{E}^{N}$ defined for any $r>0$:

$U_{r}(x_{0}^{(k)})=\pi_{k}^{-1}(B_{r}(x_{0}^{(k)}))\subset\mathbb{E}^{N}.$

As our correspondence for each $r>0$, we take

$C=\\{(\pi_{k}(w),\pi_{\infty}(w)):w\in U_{r}(x_{0}^{(k)})\\}\subset
B_{r}(x_{0}^{(k)})\times B_{r}(x_{0}^{(\infty)}).$

This is a correspondence because both smocking maps are surjections. Next,
note that

$|d_{X_{k}}(\pi_{k}(v),\pi_{k}(w))-d_{X_{\infty}}(\pi_{\infty}(v),\pi_{\infty}(w))|=|\overline{d}_{k}(v,w)-\overline{d}_{\infty}(v,w)|.$

The collection of sets $S_{\infty}$ is a Hausdorff limit, so it is the
collection of sets of limits of all converging sequences $\\{s_{k}\\}$ in
$\mathbb{E}^{N}$ such that $s_{k}\in S_{k}$ for all $k$. In particular, such
sequences can only grow with increasing $k$ with at most exponential speed, so
the distances measured by the respective smocking pseudometrics can only decay
with at most exponential speed. This implies that we have for the difference
of the distances

$|\overline{d}_{k}(v,w)-\overline{d}_{\infty}(v,w)|\leq Ae^{-f(k)},$

where $A$ is some sufficiently large constant and $f(k)$ is a monotonically
increasing function of $k$. On the other hand, such sequences could instead
decay with at most exponential speed, and the same conclusion holds after
taking the magnitude of the difference. By Theorem 7.3 of [2] for almost
distance preserving correspondences, this implies that

$d_{GH}\big{(}\big{(}B_{r}(x_{0}^{(k)}),d_{X_{k}}\big{)},\big{(}B_{r}(x_{0}^{(\infty)}),d_{X_{\infty}}\big{)}\big{)}\leq
2Ae^{-f(k)}.$

Taking the limit, we have

$\lim_{k\rightarrow\infty}d_{GH}\big{(}\big{(}B_{r}(x_{0}^{(k)}),d_{X_{k}}\big{)},\big{(}B_{r}(x_{0}^{(\infty)}),d_{X_{\infty}}\big{)}\big{)}=0.$

This completes the proof, with no need to pass to a subsequence and no
dependence on the choice of base point. $\hfill\square$

It is possible to think of potential counterexamples which are something as
follows: one imagines a smocking pattern given by alternating balls of radius
$1/k$ with balls of radius $(2^{k}-2)/2^{k}$. In the case of $k=1$, this
implies that $S_{1}$ is a pattern which alternates between unit $N$-balls and
points. In the limit as $k$ goes to infinity, $S_{k}$ will become the opposite
pattern, since the unit balls in $S_{1}$ will all shrink to points and the
points will expand out to be unit balls. This will give Hausdorff convergence
to a smocking set $S_{\infty}$ which can be smocked to obtain a well-defined
smocked space $X_{\infty}$.

However, $(2^{k}-2)/2^{k}$ approaches $1$ faster than $1/k$ approaches zero as
$k$ goes to infinity. It follows that if the balls in $S_{1}$ are sufficiently
close to the points, at some stage in the sequence of smocking sets $S_{k}$
the growing ball will merge with the shrinking ball to form a new set which
might develop such that the difference between the distance between two points
in $X_{k}$ and the distance between the corresponding two points in $X_{k+1}$
at some stage momentarily gets larger, whereas one would expect the difference
between distances between pairs of points to decrease with increasing $k$ if
there is to be Gromov-Hausdorff convergence of the metric spaces as $k$ tends
to infinity. This fails to be a counterexample because by hypothesis the
family of subsets over $\mathbb{E}^{N}$ need to remain disjoint from each
other as the sequence $\\{S_{k}\\}_{k=1}^{\infty}$ converges to its Hausdorff
limit.

##  3 Subsequential Gromov-Hausdorff Convergence of Smocked Spaces

The next question regarding Gromov-Hausdorff convergence is when a sequence of
smocked spaces $\\{X_{k}\\}_{k=1}^{\infty}$ has a Gromov-Hausdorff converging
subsequence. Gromov’s compactness theorem tells us when a sequence of metric
spaces has a such a subsequence, but the theorem requires a uniform upper
bound on diameter and the number of balls of any given radius (intuitively,
one should be able to fill the space with not too many small disjoint balls as
one goes to smaller and smaller scales) [6]. Smocked spaces have unbounded
diameter and do not satisfy these hypotheses, and so this is an open question
for smocked spaces.

In [2], it is shown that sequences of re-scalings of smocked spaces do
converge even though the spaces themselves do not satisfy the hypotheses of
the Gromov compactness theorem. There are some immediate examples of Gromov-
Hausdorff converging subsequences which one can construct. For example, take a
sequence of smocked spaces $\\{X_{k}\\}_{k=1}^{\infty}$ such that $X_{k}$ is a
smocked space with one single smocking stitch for odd $k$ and such that
$X_{k}$ is a smocked space with two smocking stitches where the distance
between the stitches in the pseudometric remains constant for even $k$. This
sequence of smocked spaces clearly contains a Gromov-Hausdorff converging
subsequence given by $(X_{2k})$.

Generally speaking, we would like conditions for a sequence to contain at
least one Gromov-Hausdorff converging subsequence and furthermore we would
like a limit space (the trivial example above has no limit space because the
$X_{k}$’s do not Gromov-Hausdorff converge). It turns out that to get such
conditions it is sufficient to assume uniform bounds on the smocking
constants. Intuitively, the result says that one cannot pack in too many sets
too closely together in the smocked space. This may explain why the examples
found by Sormani et al. can be shown to have rescaled sequences which converge
even though smocked spaces themselves do not meet the hypotheses of Gromov’s
compactness theorem.

Proposition 3.1: A sequence of smocked spaces $\\{X_{k}\\}_{k=1}^{\infty}$
contains a Gromov-Hausdorff converging subsequence if and only if there are
uniform upper bounds on the smocking constants.

Proof: The proof is similar to others in the literature, but we will reproduce
the details here for completeness [1]. We will begin by repeating the
definitions of the smocking constants [2].

Definition 3.2 The smocking depth $h$ is defined as

$h=\text{inf}\\{r:\forall x\in X\>\exists j\in J\>\exists z\in
I_{j}\>\text{such that}\>|x-z|<r\\}.$

Definition 3.3 The smocking lengths $L_{\text{max}}$ and $L_{\text{min}}$ are
defined using the diameters of the smocking stitches as follows:

$L_{\text{min}}=\text{inf}\\{D(I_{j}):\>j\in J\\},$
$L_{\text{max}}=\text{sup}\\{D(I_{j}):\>j\in J\\}.$

The fourth constant is the smocking separation factor $\delta$, which was
defined in the Introduction. We will also recall the definition of an
$\epsilon$-net.

Definition 3.4 An $\epsilon$-net in a pointed metric space $(X,x)$ is a subset
$L$ of $X$ containing $x$ whose $\epsilon$-neighbourhood covers $X$ and for
which there is some $\epsilon^{\prime}>0$ with
$d(l_{1},l_{2})\geq\epsilon^{\prime}$ for all $l_{1}\neq l_{2}$ in $L$.

We must prove that there is a finite number of indices $i_{1},...,i_{p}$ such
that for every $k$, there exists $j$ with

$d_{GH}((B_{r}(x_{0}^{(k)}),dX_{k}),(B_{r}(x_{0}^{(i_{j})}),dX_{i_{j}}))\leq\text{constant}\cdot
D_{k},$

where $D_{k}$ is defined to be
$\text{sup}\\{L_{\text{min}}^{k},L_{\text{max}}^{k},h_{k},\delta_{k}\\}$.
$D_{k}$ is assumed to be finite by hypothesis.

Since $D_{k}$ is finite, for each ball $(B_{r}(x_{0}^{(k)}),dX_{k})$ the
maximal disjoint filling is achieved by balls of radius $cD_{k}$ centred at
points of a $2cD_{k}$-net of the ball (call the net $L$), for some $c>0$. The
number of balls in the filling is bounded by some quantity
$N=N(D_{k},x_{0}^{(k)},r)$. A choice of bijection
$f:\\{1,...,N\\}\longrightarrow L$ induces a distance function $d_{L}$ on the
set $\\{1,...,N\\}$. The function
$d_{L}:\\{1,...,N\\}\times\\{1,...,N\\}\longrightarrow\mathbb{R}_{+}$ which
assumes values between $2cD_{k}$ and $2r$ on the subset of $S$ of pairs of
different integers has $N^{2}-N$ elements. The space $[2cD_{k},2r]^{N^{2}-N}$
is precompact with respect to the product distance function, so there exist
finitely many indices $i_{1},...,i_{p}$ such that, for each $i$, there exists
$j$ such that

$\sup_{S}|d_{k}(m,q)-d_{i_{j}}(m,q)|\leq c^{2}D_{k}^{2},$

which implies that

$\sup_{S}\frac{d_{k}(m,q)}{d_{i_{j}}(m,q)}\leq\frac{1+cD_{k}}{1-cD_{k}}.$

This in turn implies that

$d_{H}((B_{r}(x_{0}^{(k)}),dX_{k}),(B_{r}(x_{0}^{(i_{j})}),dX_{i_{j}}))\leq(2r+4)cD_{k}.$

Finally, Hausdorff convergence implies Gromov-Hausdorff convergence, so we
have

$d_{GH}((B_{r}(x_{0}^{(k)}),dX_{k}),(B_{r}(x_{0}^{(i_{j})}),dX_{i_{j}}))\leq(2r+4)cD_{k},$

as required. For the other direction of the implication, it is clear that
containing a well-defined Gromov-Hausdorff limit of subsequences such as a re-
scaling implies that the smocking constants must have a uniform bound on them
from above (ie. none of the sets in $S_{k}$ can be infinitely long or
infinitely far apart), which completes the proof. $\hfill\square$

##  4 Measured Convergence of Smocked Measure Spaces

We would now like to consider the results which can be obtained when we extend
to metric spaces which are also equipped with measures. We will begin by
recalling the definition of metric measure spaces given in [7].

Definition 4.1 A metric measure space is a triple $(X,d,\mu)$, where $(X,d)$
is a metric space and $\mu$ is a measure on the Borel $\sigma$-algebra of $X$.

We then propose to extend this definition to smocked spaces in order to work
with what we will call pointed smocked metric measure spaces (or smocked
measure spaces for short). As before, we must use the pointed condition
because smocked spaces are non-compact.

Definition 4.2 A smocked metric measure space is a triple $(X,d,\mu)$, where
$(X,d)$ is a smocked space and $\mu$ is a non-zero Borel measure which is
finite on bounded sets, and a pointed smocked metric measure space
$(X,d,\mu,\overline{x})$ is a smocked metric measured space together with a
given point $\overline{x}\in\text{supp}(\mu)\subset X$.

The first question we might ask is whether we can obtain analogues of our
previous convergence results established above, but now for the case of
smocked measure spaces. We will begin by defining a suitable notion of pointed
measured Gromov-Hausdorff convergence used by Lott and Villani and adapted by
Gigli, Mondino and Savaré [8-10]. This notion works with proper (that is,
bounded closed sets are compact) pointed metric measure spaces.
$\mathbb{E}^{N}$ is proper as well as being finite-dimensional, so distances
will always converge uniformly on bounded sets.

Definition 4.3 If one has a sequence of pointed smocked metric measure spaces
$(X_{j},d_{j},\mu_{j},\overline{x}_{j})$, pointed measured Gromov-Hausdorff
convergence is defined as follows:

$(X_{j},d_{j},\mu_{j},\overline{x}_{j})\xrightarrow{pmGH}(X_{\infty},d_{\infty},\mu_{\infty},\overline{x}_{\infty})$

if and only if for every $\epsilon,R>0$ there exists $N(\epsilon,R)$ such
that, for all $n\geq N$, there exists a Borel map $f_{j}^{R,\epsilon}$:
$B_{R}(\overline{x}_{j})\rightarrow\overline{x}_{\infty}$ such that

(i) $f_{j}^{R,\epsilon}(\overline{x}_{j})=\overline{x}_{\infty}$;
(ii) $\text{sup}_{x,y\in
B_{R}(\overline{x}_{j})}|d_{j}(x,y)-d_{\infty}(f_{j}^{R,\epsilon}(x),(f_{j}^{R,\epsilon}(y))|\leq\epsilon$;
(iii) the $\epsilon$-neighbourhood of
$f_{j}^{R,\epsilon}(B_{R}(\overline{x}_{j}))$ contains
$B_{R-\epsilon}(\overline{x}_{\infty})$;
(iv) $(f_{j}^{R,\epsilon})_{\\#}(\mu_{j}\rvert_{B_{R}(\overline{x}_{j})})$
weakly converges to $\mu_{\infty}\rvert_{B_{R}(\overline{x}_{\infty})}$ as
$n\rightarrow\infty$, for almost every $R>0$.

This definition is somewhat more complicated than the one for pointed Gromov-
Hausdorff convergence, but we can still show that if one has a sequence of
smocking sets $S_{k}$ which Hausdorff converge to $S_{\infty}$ as before, then
we can always define corresponding smocked measure spaces $X_{k}$ which
converge to a pointed measured Gromov-Hausdorff limit. By Proposition 2.1, it
is only necessary to show that pointed Gromov-Hausdorff convergence implies
pointed measured Gromov-Hausdorff convergence.

Proposition 4.4: If a sequence of smocked spaces $(X_{k},x_{k},d_{k})$
converges in the pointed Gromov-Hausdorff sense to
$(X_{\infty},x_{\infty},d_{\infty})$, then one can define a sequence
$(X_{k},d_{k},\mu_{k},\overline{x}_{k})$ which converges in the pointed
measured Gromov-Hausdorff sense to
$(X_{\infty},d_{\infty},\mu_{\infty},\overline{x}_{\infty})$.

Proof: Since we already have pointed Gromov-Hausdorff convergence of
$(X_{k},x_{k},d_{k})$ to $(X_{\infty},x_{\infty},d_{\infty})$, the Borel maps
in question can be chosen essentially arbitrarily, much as in the proof of
Proposition 3.28 of [9]. For $k\in\mathbb{N}$, start with two sequences
$\overline{R}_{k}$ and $\overline{\epsilon}_{k}$ such that
$\overline{R}_{k}\rightarrow\infty$ and
$\overline{\epsilon}_{k}\rightarrow\infty$ as $k\rightarrow\infty$. Set
$N_{k}=N(\overline{\epsilon}_{k},\overline{R}_{k})$ and define two sequences
$R_{j}\rightarrow\infty$ and $\epsilon_{j}\rightarrow 0$ such that $R_{j}$ and
$\epsilon_{j}$ are arbitrary when $j<N_{1}$, whereas for $N_{k}\leq j\leq
N_{k+1}$, set $R_{j}=\overline{R}_{k}$ and
$\epsilon_{j}=\overline{\epsilon}_{k}$. Next, define Borel maps
$f_{j}:X_{j}\rightarrow X_{\infty}$ such that $f_{j}$ is arbitrary for
$j<N_{1}$ and $f_{j}$ is arbitrary on the complement of
$B_{R_{j}}(\overline{x}_{j})$ when $N_{k}\leq j\leq N_{k+1}$. On
$B_{R_{j}}(\overline{x}_{j})$ itself, the map $f_{j}$ is set to be the Borel
map $f_{j}^{R_{j},\epsilon_{j}}$. One can then re-define the arbitrary
sequences $R_{j}$ and $\epsilon_{j}$ and the maps $f_{j}$ for $j<N_{1}$ as
necessary to achieve each of the sub-implications in Definition 4.3 (where the
same Borel map is used for each one). Since $(f_{n})$ can be re-defined as
needed, there is no dependence on the the choice of base point
$\overline{x}_{j}\in\text{supp}(\mu_{j})\subset X_{j}$. $\hfill\square$

The following corollary is immediate.

Corollary 4.5: If one defines a sequence of smocked measure spaces using
smocking sets $S_{k}\subset\mathbb{E}^{N}$ such that the sets Hausdorff
converge in the limit to a smocking set $S_{\infty}$, then one can define
corresponding smocked measure spaces which converge to the corresponding limit
space in the pointed measured Gromov-Hausdorff sense.

We will next consider if there is some suitable pre-compactness result in the
case of pointed measured Gromov-Hausdorff convergence. Note that the notion of
measured Gromov-Hausdorff convergence which we are using can be rephrased so
that it is equivalent to convergence in the Gromov-Hausdorff-Prokhorov
topology studied in [11, 12] when specialised to locally compact length spaces
endowed with locally finite measures.

Proposition 4.6: A sequence of smocked measure spaces
$\\{(X_{k},d_{k},\mu_{k},\overline{x}_{k})\\}_{k=1}^{\infty}$ contains a
subsequence which converges in the measured Gromov-Hausdorff topology if and
only if there is a uniform upper bound on the smocking constants and
$\text{sup}\>\mu_{k}(B_{r}(x_{0}^{(k)}))<\infty$.

Proof: One way to prove this would be the usual approach of explicitly
constructing an abstract limit space with an ancillary measure and showing
that a convergence can be obtained after extraction of subsequences [1]. The
proof can be done be following the proof of Thm 2.11 in [11] with some minor
modifications, so we will not reproduce the details here. $\hfill\square$

##  5 Smocked Wasserstein Spaces

We have shown some basic convergence results for smocked measure spaces and we
would now like to establish whether one can get any results for stability
under convergence when working with this type of metric measure space. The
most obvious thing to start with in the context of previous work on optimal
transport would the curvature dimension condition (denoted by
$\text{CD}(K,N)$), which morally means that the Ricci curvature is bounded
from below by $K\in\mathbb{R}$ and the dimension is bounded from above by a
real number $N$ which is not less than $1$ [13, 14]. The connection between
lower bounds on the Ricci curvature and convergence of metric measure spaces
has a rich history, especially in connection with the volume convergence
conjecture of Anderson and Cheeger [15-17]. Lott and Villani showed that the
$\text{CD}(K,N)$ condition is stable under pointed measured Gromov-Hausdorff
convergence for proper pointed metric measure spaces (as long as the spaces
are also compact length spaces) [10]. Gigli, Mondino and Savaré developed a
slightly different notion of convergence (called pointed measured Gromov
convergence) which does not require compactness assumptions on the metric
structure and showed that the $\text{CD}(K,\infty)$ condition is stable under
this type of convergence [9].

As usual, our approach will be via a metric space called the Wasserstein space
which is canonically associated to a metric space [14]:

Definition 5.1: The Wasserstein space $\mathcal{P}_{p}(X):=(P(X),W_{p})$ is a
metric space associated to $X$, where $P(X)$ is the space of Borel probability
measures on $X$ and $W_{p}$ is the Wasserstein distance of order $p$ between
two measures $\mu$ and $\nu$, which is defined to be

$W_{p}(\mu,\nu)=\Bigg{(}\inf_{\pi}\int_{X\times
X}d(x,y)^{p}d\pi(x,y)\Bigg{)}^{1/p},$

where $\pi$ can take values in the set of all transference plans between $\mu$
and $\nu$.

Definition 5.2 ($\text{CD}(K,N)$ space): Let $K\in\mathbb{R}$. A (pointed)
metric measure space is said to have Ricci curvature which is bounded from
below by $K$ if for any pair of measures
$\mu_{1},\mu_{2}\in\mathcal{P}_{2}(X)$ with finite relative entropy, there
exists a $W_{2}$-geodesic $\Gamma:[0,1]\rightarrow\mathcal{P}_{2}(X)$
connecting the two measures such that

$\text{Ent}(\Gamma(t)|\mu)\leq(1-t)\text{Ent}(\Gamma(0)|\mu)+t(\Gamma(1)|\mu)-\frac{K}{2}t(1-t)W_{2}^{2}(\Gamma(0),\Gamma(1)),$

for all $t\in[0,1]$, where the relative entropy functional on
$\mathcal{P}_{2}(X)$ is defined to be

$\text{Ent}(\mu_{1}|\mu)=\int_{X}\frac{d\mu_{1}}{d\mu}\log\bigg{(}\frac{d\mu_{1}}{d\mu}\bigg{)}d\mu.$

Unfortunately, there is no stability result to be found here for measured
convergence of smocked measure spaces, because the curvature dimension
condition never holds for smocked spaces and one cannot bound the synthetic
Ricci curvature from below. This is because smocked spaces are not themselves
length spaces or compact spaces, so that immediately excludes the possibility
that $\mathcal{P}_{2}(X)$ could be a length space or a compact space (see
Proposition 2.10 of [7], for example). Since $\mathcal{P}_{2}(X)$ is not a
length space, it does not have geodesics.

However, we do have Gromov-Hausdorff convergence of Wasserstein spaces
associated to a sequence of smocked spaces by generalisation of existing
results of Lott and Villani to pointed (not necessarily compact) metric
spaces.

Proposition 5.3: If one defines a sequence of smocked spaces
$\\{X_{k}\\}_{k=1}^{\infty}$ using a sequence of smocking sets
$\\{S_{k}\\}_{k=1}^{\infty}$ such that the sets converge to a set $S_{\infty}$
in the Hausdorff topology, then the sequence of associated Wasserstein spaces
$\\{\mathcal{P}_{2}(X_{k})\\}_{k=1}^{\infty}$ Gromov-Hausdorff converges to
the corresponding Wasserstein space $\mathcal{P}_{2}(X_{\infty})$ associated
to the smocked space $X_{\infty}$ defined using the Hausdorff limit
$S_{\infty}$.

Proof: By the methods in Appendix E of [10], one can extend Proposition 4.1 of
[10] to non-compact metric spaces by defining the approximations used to balls
in $X_{k}$, rather than the whole of $X_{k}$. This implies that if a sequence
of pointed metric spaces $\\{(X_{k},x_{k},d_{k})\\}_{k=1}^{\infty}$ converges
in the pointed Gromov-Hausdorff topology to a pointed metric space
$(X_{\infty},x_{\infty},d_{\infty})$, then
$\\{\mathcal{P}_{2}(X_{k})\\}_{k=1}^{\infty}$ converges in the Gromov-
Hausdorff topology to $\mathcal{P}_{2}(X_{\infty})$. Proposition 2.1 then
implies that if a sequence of smocked spaces $\\{X_{k}\\}_{k=1}^{\infty}$ is
defined using a sequence of smocking sets $\\{(S_{k})_{k=1}^{\infty}\\}$ such
that the sets Hausdorff converge to a set $S_{\infty}$, the associated
Wasserstein spaces Gromov-Hausdorff converge to $\mathcal{P}_{2}(X_{\infty})$.
$\hfill\square$

Pulled thread spaces (and more generally smocked spaces) can be defined by
starting with any geodesic metric space or length space [1]. This implies that
the Wasserstein spaces studied in [9] can be smocked themselves by extending
all the definitions in [2] in the analogous way, since the spaces they use are
associated to length spaces and so must themselves be length spaces. This
opens up the possibility of having a smocked space which does not have an
obvious geometric interpretation.

Definition 5.4: A smocked Wasserstein space $X$ is a smocked space which is
defined using a Wasserstein space $\mathcal{P}_{p}(X)$ and smocking stitches
which are disjoint connected compact subsets of the space of Borel probability
measures $P(X)$.

As emphasised before, not all Wasserstein spaces can be smocked as they are
not all length spaces (in particular, the Wasserstein spaces associated to
smocked spaces cannot be smocked). Intuitively, the usual notion of the
Wasserstein distance being the minimal cost to transport mass from one measure
to another still holds for a smocked Wasserstein space, but now there are
somehow areas in the space of measures which act like wormholes and have no
cost attached to them when used to transport mass. We will define here the
Gromov-$W_{p}$-Wasserstein distance between metric measure spaces [9]:

Definition 5.5: Given two metric measure spaces $(X_{1},d_{1},\mu_{1})$ and
$(X_{2},d_{2},\mu_{2})$, the Gromov-$W_{p}$-Wasserstein distance for any
$p\geq 1$ is defined as

$d_{GW_{p}}((X_{1},d_{1},\mu_{1}),(X_{2},d_{2},\mu_{2})):=\inf
d_{W_{P}}^{Z}((\phi_{1})_{*}\mu_{1},(\phi_{2})_{*}\mu_{2}),$

where the infimum is taken over all isometric embeddings
$\phi_{i}:(X_{i},d_{i})\rightarrow(Z,d_{Z})$ into a common metric space
$(Z,d_{Z})$.

This distance generalises the D-distance of Sturm to all exponents $p$, where
the relation of the Sturm distance to the Wasserstein distance is obviously
analogous to the relation which the Gromov-Hausdorff distance has to the
Hausdorff distance [7]. It can be shown that $d_{GW_{p}}$ satisfies all the
properties of a pseudometric which is allowed to have the value of $\infty$
and that the Gromov-$W_{p}$-Wasserstein distance between two metric measure
spaces is zero if and only if the two spaces have isometric support [18].

One question we can ask here is whether and how smocking the Wasserstein
spaces has an influence on $GW_{p}$-convergence of the original metric measure
spaces (the original spaces being compact or non-compact length spaces). We
will accordingly define a smocked $GW_{p}$ distance (which we may denote by
$sGW_{p}$ for brevity).

Definition 5.6: Given two metric measure spaces $(X_{1},d_{1},\mu_{1})$ and
$(X_{2},d_{2},\mu_{2})$, the smocked Gromov-$W_{p}$-Wasserstein distance for
any $p\geq 1$ is defined as

$d_{sGW_{p}}((X_{1},d_{1},\mu_{1}),(X_{2},d_{2},\mu_{2})):=\inf
d_{sW_{P}}^{Z}((\phi_{1})_{*}\mu_{1},(\phi_{2})_{*}\mu_{2}),$

where the infimum is taken over all isometric embeddings as before and
$d_{sWP}$ denotes a smocked Wasserstein distance on a space of measures.

Intuitively, one might guess that there is always some smocking such that
convergence to a limit space in the smocked $d_{GW_{p}}$ metric occurs even
when there might not be convergence in the usual $d_{GW_{p}}$ metric (hence
$d_{sGW_{p}}$ convergence is weaker than $d_{GW_{p}}$ convergence):

$\lim_{k\rightarrow\infty}d_{sGW_{p}}\big{(}\big{(}X_{k},d_{X_{k}},\mu_{k}\big{)},\big{(}X_{\infty},d_{X_{\infty}},\mu_{\infty}\big{)}\big{)}=0.$

This expectation arises from the fact that adding a smocking set and a
smocking map $\pi:\mathcal{P}_{p}(X)\rightarrow X$ to a Wasserstein space can
be done in such a way that a pair of measures can be made to be arbitrarily
close to one another.

Proposition 5.7: There exists a limit space
$(X_{\infty},d_{X_{\infty}},\mu_{\infty})$ to which the sequence
$\\{(X_{k},d_{X_{k}},\mu_{k})\\}_{k=1}^{\infty}$ converges in the smocked
$d_{GW_{p}}$ metric, where $X_{k}$ and $X_{\infty}$ are compact length spaces.

Proof: The smocking set and smocking map on the Wasserstein space can be
chosen so that one has

$d_{sW_{p}}(\mu_{k}(X_{k},d_{X_{k}}),\mu_{\infty}(X_{\infty},d_{X_{\infty}}))\leq\epsilon_{n},$

where $\epsilon_{n}$ is a sequence which converges to zero for all
$n\in\mathbb{N}$ and infimums are taken over all metrics on the disjoint union
of $X_{k}$ and $X_{\infty}$ such that the embeddings $X_{k}\rightarrow
X_{k}\amalg X_{\infty}$ and $X_{\infty}\rightarrow X_{k}\amalg X_{\infty}$ are
isometries. However, Wasserstein convergence implies Gromov-Wasserstein
convergence, so that we have

$\lim_{k\rightarrow\infty}d_{sGW_{p}}\big{(}\big{(}X_{k},d_{X_{k}},\mu_{k}\big{)},\big{(}X_{\infty},d_{X_{\infty}},\mu_{\infty}\big{)}\big{)}=0$

as required. $\hfill\square$

Note that although Gromov-$W_{1}$-Wasserstein convergence implies pointed
measured Gromov convergence, the latter does not necessarily imply pointed
measured Gromov-Hausdorff convergence except for the special case of doubling
spaces [9].

Another basic principle of optimal transport is that a pair of Borel
probability measures on a manifold which evolve under a diffusion equation
should converge in the Wasserstein distance if the manifold satisfies a
curvature condition (usually positive Ricci curvature as defined above) [19].
The work of Sturm and von Renesse shows that if a smooth, compact Riemannian
manifold $M$ is equipped with two Borel probability measures $\mu$ and $\nu$
which spread according to the heat equation, then the lower bound on the Ricci
curvature of $M$ controls the speed with which the Wasserstein distance
between the two measures can expand or decay over time. In particular, it was
shown that the following properties are all equivalent for such a manifold:

Proposition 5.8 [20]: For $K\in\mathbb{R}$ and a compact Riemannian manifold
$M$, one has

(i) $\text{Ric}(M)\geq K$.
(ii) For all $x$ and $y$ in $M$ and all $t>0$, there exists $r\in[1,\infty]$
with

$d_{W_{P}}(p_{t}(x,\cdot),p_{t}(y,\cdot))\leq e^{-Kt}\cdot d(x,y).$

(iii) For all $r\in[1.\infty]$, all $\mu$ and $\nu$ in $\mathcal{P}_{P}(M)$,
and all $t>0$

$d_{W_{P}}(\mu p_{t}(x,\cdot),\nu p_{t}(y,\cdot))\leq e^{-Kt}\cdot
d_{W_{P}}(\mu,\nu),$

where $p_{t}(x,y)$ denotes the minimal positive fundamental solution of the
heat equation.

Again, since the smocking set and smocking map can be chosen arbitrarily,
properties (ii) and (iii) can always be made to hold when the smocked
$d_{W_{P}}$ metric is used. It may be possible to show that these properties
still imply the same Ricci curvature bound with certain cases of the
$d_{sW_{p}}$ metric (for example, when the smocking is periodic). This would
be interesting, since the $d_{sW_{P}}$ metric is a weaker distance notion than
the usual $d_{W_{P}}$ metric.

##  6 Gromov-Hausdorff Convergence of Smocked Wasserstein Spaces

One of the main reasons for the introduction of smocked metric spaces by
Sormani et al. is that they can have pointed Gromov-Hausdorff limit spaces
with interesting properties [2]. We would next like to know if one can obtain
results on Gromov-Hausdorff convergence of smocked Wasserstein-type metrics.
Note that this differs from the Gromov-Hausdorff convergence of the
Wasserstein spaces in Proposition 5.3, since those Wasserstein spaces were
associated to smocked spaces and not smocked themselves, whereas the
Wasserstein spaces we discuss here are smocked. Convergence of Wasserstein-
type transport distances for Borel probability measures on discrete sets was
previously studied in [21].

A recent circle of ideas has started to investigate the question of whether
the discrete Wasserstein space on a random point cloud on the flat torus
$\mathbb{T}^{d}:=\mathbb{R}^{d}/\mathbb{Z}^{d}$ approximates Wasserstein space
in the continuum. Trillos proved that for a uniform distribution of points on
$\mathbb{T}^{d}$, discrete analogues of the Wasserstein metric on random
geometric graphs converge in the Gromov-Hausdorff sense to the usual continuum
Wasserstein distance [21]. Furthermore, there is a clear correspondence
between the continuum $d$-torus and the discrete $d$-torus
$\mathbb{T}^{d}_{N}$ with mesh size $1/N$ such that one can show that the
discrete transport metric on $\mathbb{T}^{d}_{N}$ converges to the standard
Wasserstein distance on $\mathbb{T}^{d}$ in the limit as $N$ goes to infinity
[22].

However, it is possible to introduce periodicity without working on the torus
because smocking patterns on $\mathbb{R}^{d}$ can be periodic. Beginning with
the Wasserstein space $\mathcal{P}_{2}(\mathbb{R}^{d})$, this space can be
equipped with the Wasserstein distance induced by the Euclidean distance on
$\mathbb{R}^{d}$. The distance between two measures is defined to be

$W_{2}(\mu,\nu)=\min_{\pi\in\Gamma(\mu,\nu)}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{2}\>d\pi(x,y),$

where $\mu$, $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\Gamma(\mu,\nu)$
denotes the set of transference plans from $\mu$ into $\nu$ and $|x-y|$ is the
Euclidean distance between two points $x$ and $y$ in $\mathbb{R}^{d}$.
$\mathcal{P}_{2}(\mathbb{R}^{d})$ is a length space and so can be smocked.

By the work of Benamou and Brenier, the Wasserstein distance between two
measures is connected by a curve in $\mathcal{P}_{2}(\mathbb{R}^{d})$ which
satisfies a stationary action principle [23]:

$W_{2}(\mu,\nu)=\min\int^{1}_{0}\int_{\mathbb{R}^{d}}|V_{t}(x)|^{2}\>d\mu_{t}\>dt,$

where the minimum is taken over all pairs $(\mu_{t},V_{t})$ which solve the
usual continuity equation of fluid dynamics

$\frac{d}{dt}\mu_{t}+\text{div}(V_{t}\mu_{t})=0,$

.

where $\mu_{0}=\mu$ and $\mu_{1}=\nu$.

Proposition 6.1: If all the sets of measures in the associated smocking
patterns decay to sets each composed of one single measure with speed $1/k$ as
$k$ goes to infinity, then the smocked metric spaces
$(\mathcal{P}(\mathbb{R}^{d}_{k}),sW_{2}^{(k)})$ converge to the continuum
space $(\mathcal{P}(\mathbb{R}^{d}),W_{2})$ in the Gromov-Hausdorff sense.

Proof: This follows again from Proposition 2.1. If all the smocking subsets of
$\mathcal{P}(\mathbb{R}^{d})$ converge to points with speed $1/k$, then the
smocking set converges to a collection of isolated measures in the Hausdorff
sense as $k$ goes to infinity. Hausdorff convergence of the smocking set
implies Gromov-Hausdorff convergence of the associated Wasserstein spaces, so
$\\{(\mathcal{P}(\mathbb{R}^{d}_{k}),sW_{k})\\}_{k=1}^{\infty}$ converges to
$(\mathcal{P}(\mathbb{R}^{d}),W_{2}^{(k)})$ in the Gromov-Hausdorff sense.
$\hfill\square$

We may refer to a sequence of smocking sets which converges with this speed to
a collection of points a sequence of shrinking smocking sets. This by itself
is not particular useful. However, we will make a definition which motivates
this convergence result.

Definition 6.2: A smocking set is said to be $\epsilon$-regular if each subset
in the smocking set is at a distance of $\epsilon$ from a neighbouring subset
and if each subset has the same shape.

Applied to the examples of [2], the smocking sets used for the ribbed smocked
space, the flower smocked space and the bumpy smocked space are all
$\epsilon$-regular. The terminology is meant to be evocative of regular
tilings, which are tilings composed of tiles of the same shape.

If a smocking pattern used in the space
$(\mathcal{P}(\mathbb{R}^{d}_{k}),sW_{2}^{(k)})$ is $\epsilon$-regular and
periodic, then one can replace the discrete Wasserstein distance on the
discrete $d$-torus induced by the kernel for a simple periodic random walk
with an equivalent periodic smocked Wasserstein distance on $\mathbb{R}^{d}$
itself. The main advantage which might be conveyed by this is that the
analytic issues involved in showing convergence of a discrete transport metric
on a discrete $d$-torus to the Wasserstein metric on the continuous torus are
fairly involved [21]. In particular, whether or not an evolutionary PDE
defined on a discrete Wasserstein space is consistent or whether it evolves
towards the continuous analogue PDE defined on the original manifold is
unknown, even if the discrete Wasserstein spaces do converge to the continuous
Wasserstein space [21, 24].

On the other hand, defining evolutionary PDEs on a sequence of smocked
Wasserstein spaces might be expected to have fewer analytic subtleties.
Depending on the smocking in question it may be easier to consider sequences
of smocked Wasserstein spaces, knowing that the spaces must eventually
converge to the proper continuous Wasserstein space for $\mathbb{R}^{d}$ by
Proposition 6.1 (as long as the smocking sets shrink in the limit). It is also
worth mentioning that the approach with smocked Wasserstein spaces means that
one can work with $\mathbb{R}^{d}$ as the underlying manifold, rather than
$\mathbb{R}^{d}/\mathbb{Z}^{d}$.

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