j: all relatives j who had not developed diabetes by the subject’s examination
date

Kx: percent of genes shared by relative x and set at:

  * •

0.5 when the relative x is a parent or full sibling

  * •

0.25 when the relative x is a half sibling, grandparent, aunt or uncle

  * •

0.125 when the relative x is a half aunt, half uncle or cousin

ADMi: age in years of relative i when diabetes was diagnosed

ACLj: age in years of relative at the last non-diabetic examination

88 / 14: maximum and minimum ages at which relatives developed DM

20 / 50: moderating constants

This definition provides semantical and structural data requirements for the
DPF variable thoroughly embedded into the domain of diabetes research
(contextual data requirements). This definition could be developed into formal
representations that support re-use and merging this dataset in other ML model
training. Similar domain knowledge exists for glucose, blood pressure, skin
thickness, and insulin.

Model training.

Model training is governed by performance requirements as constraints for
functional requirements. For building trust in medical treatment, prediction
accuracy should be high with a high sensitivity (recall) value close to
100$\%$; i.e., the percentage of missed positives should be small. Slightly
less important is specificity (true negative rate); that is, not many people
should receive treatment, even though they are healthy. These tradeoffs and
thresholds must be based on domain knowledge. Reported sensitivity (0.78) and
specificity scores (0.77) are taken as reference values.

At the beginning, a series of machine learning model types is trained and
evaluated by applying default hyperparameter values. Accordingly, Gradient
boosting performs best with an accuracy of more than 88$\%$, recall
(sensitivity) 87$\%$, precision 88$\%$ and F1 value 88$\%$ [125]. Model
training depends on selecting the best ML model type and leveraging domain
knowledge. Selecting the best ML model type is the core of any model training
phase, and requires application of different models and validation of model
performance. Leveraging domain knowledge depends on the interaction between
domain experts and data scientists. Domain knowledge leveraging supports:
knowledge to data or data to knowledge. The first direction, knowledge to
data, is used if a domain expert can express domain knowledge in a way that
can be transformed into additional features. For instance, if a person is
younger than 30 years with a plasma glucose level <120 mg/dl, then she is less
likely to suffer from diabetes in the next 5 years. This heuristic rule can be
expressed by an OCL rule with an added binary feature $kl1$ initialized with
$0$ for all samples:

context Person
inv: self.Patient.age < 30 and
self.medical_entry.glucose < 120
-> forAll(e $|$ e.kl1 = 1)

For data to knowledge, a data scientist analyzes the dataset and attempts to
extract heuristic rules that will subsequently be evaluated by domain experts.
For instance, if a data scientist finds support for a hypothetical rule that
younger women with fewer pregnancies are less likely to suffer from diabetes
in the coming years, then this is expressed as an OCL rule:

context Person
inv: self.Patient.age < 30 and
self.medical_entry.pregnancies <$=$ 6
->forAll(e $|$ e.kl2 $=$ 1)

If rule $kl2$ is verified by domain experts, another binary variable is added
to the dataset. From the data to knowledge strategy, an additional 16 binary
features were found and added to the
dataset.141414https://www.kaggle.com/vincentlugat/pima-indians-diabetes-eda-
prediction-0-906 These heuristic rules increased model performance from an
accuracy of 0.73 for gradient boosting to 0.89 with a recall of 0.84 and a
precision of 0.86. That is an increase of more than 20$\%$ above the initial
model performance.

The following ML model types were trained as shown in Table 7.

Model Accuracy AUC Recall / Sensitivity Precision F1 Light Gradient Boosting
0.89 0.94 0.84 0.86 0.85 Gradient boosting 0.89 0.95 0.81 0.85 0.83 Logistic
regression 0.84 0.91 0.73 0.78 0.76 Support vector classifier 0.85 0.91 0.75
0.81 0.78 Decision tree 0.86 0.81 0.82 0.79 0.81 K nearest neighbors 0.80 0.88
0.59 0.77 0.67

Table 7: Model types.

Model optimization.

Model optimization is also governed by performance requirements. Most machine
learning model types have hyperparameters, such as the number of neighbors$k$
that are used in KNN. Finding an optimal set of hyperparameters is a NP-
complete search problem.151515Thank you to an anonymous reviewer for this
suggestion. Even if an optimal set of hyperparameters could be computed, it is
not possible to assess whether a viable solution has been identified because
the solution might suffer from inductive fallacy. Best practices can be
expressed by heuristic rules or knowledge graphs. Model optimization is a
technical task similar to optimization of a database system by configuration
of database management parameters. More research is needed to understand the
impact on domain knowledge and on data science knowledge for optimizing ML
models.

Model integration.

After finalizing the ML model, it is integrated into the information system.
Validation procedures can be used to assess compliance with business
requirements, goal requirements, data requirements, legal requirements,
ethical requirements, and functional and non-functional requirements. Field
tests on newly collected data are used to build trust in information system
performance. Empirical studies on information systems adoption, usability,
cost effectiveness and other non-functional requirements are used for
practical evidence of ML development results. The diabetes machine learning
model was not integrated into an information system. Therefore, model
integration is not relevant in this example.

Analytical decision making.

The value of a diabetes information system lies in its potential to support
medical workers. By sampling new data, medical workers receive predictions on
the risk of patients suffering from diabetes in the future so that
countermeasures can be recommended, even in real-time.

##  5 Machine Learning for Conceptual Modeling

Machine learning can also contribute to conceptual modeling. Many of the
challenges of applying machine learning to conceptual modeling deal with
knowledge generation. Here, the term knowledge refers generally to the
constructs of a conceptual model. Knowledge challenges can be organized into
three categories: incomplete knowledge, incorrect knowledge, and inconsistent
knowledge. Incomplete knowledge includes missing or limited entities and/or
relationships. Incorrect knowledge includes incorrect entity and relationship
labels or incorrect facts (e.g., cardinalities). Inconsistent knowledge
includes different labels for the same entity or merging entities with the
same labels.

The first, and probably easiest to understand, is missing entities or
relationships. Knowledge could be extracted to identify where there is
incompleteness in modeling of an application domain, or potential missing
relationships, which would require interaction between a domain expert and a
conceptual modeling expert.

It is possible to infer what concepts and synonyms are extracted from a text.
Then, the potential entity concepts can be used to create a graph that might
indicate missing pieces or something that is incorrectly labeled (wrong entity
label recognition). There could also be inconsistent relationships, or the
potential to make incorrect inferences. Such basic kinds of research
challenges are well-known. However, anchoring such inferences in knowledge
graphs could support the combining of research on knowledge graphs with data
analytics and conceptual modeling. We can consider analytics on text and how
to extract conceptual modeling-like structures from it, as well as rule and
graph mining.

  * •

General supervised learning: linear regression, support vector machines,
decision trees, random forests, boosting models, multi-layer perceptrons, deep
neural networks [180]

  * •

Sequence learning: recurrent neural networks (incl. LSTM) [92]

  * •

Generative learning: generative adversarial neural networks [77]

  * •

Graph learning: graph neural networks, recurrent graph neural networks,
convolutional graph neural networks ([221])

  * •

Unsupervised learning: KNN, k-means, PCA [86]

  * •

Reinforcement learning: dynamic learning of agents through rewards gained from
actions in environments [200].

Extracting knowledge structures from datasets by means of machine learning is
a fast growing research field. Table 8 provides an non-exhaustive overview of
using machine learning models for extracting different conceptual model
structures. Associative rules is a robust technology for extracting relational
knowledge. Approaches for estimating relationships between entities as link
predictions are more sophisticated. Process discovery based on analysing log
files is another promising area of research. These approaches, however, do not
consider semantics. This might be why ontology extraction using machine
learning is still restricted to ontology matching and mapping, although there
are successful language translation systems that do not have explicit semantic
representations [220].

ML $\rightarrow$ CM Rules, anomalies and explanations Semantic models  (ERM,
i*) Ontologies Process models (EPC, BPMN) General Supervised learning
Associative rules [3]; rule extraction [12] Ontology mapping and matching [54,
151] Process discovery [11]; event abstraction [207] Sequence learning Rule
extraction [146] Named entity recognition [45]; link prediction [42] Ontology
matching [106] Generative learning Link prediction (Qin et al. 2020) Graph
learning Link prediction [6, 51] Relational learning [152] Unsupervised
learning Anomaly detection [5] Link prediction [122] Concept learning Mi et
al. 2020) Event abstraction [207] Reinforcement learning Rule extraction [169]

Table 8: Extraction of conceptual models with machine learning.

###  5.1 Text mining

Text mining, which discoveres conceptual structures from unstructured sources
[218], became popular with the increasing use of social media, such as
Facebook and Twitter. Various natural language processing (NLP) methods are
available for filtering keywords based on domain knowledge and domain lexica.
Preprocessing of textual data removes stopwords and reduces words to word
stems. Shallow parsing identifies phrases and recognizes named entities with
ontology mappings [21]. At a semantic level, word sense disambiguation and
identification of negations are processed. Negation is difficult to deal with
because it might mean that entities, relationships, and larger conceptual
structures are excluded, or do not exist. Entity linking by NLP methods
leverage ontologies [219]. Particularly challenging is resolving references
given by anaphora or by spatio-temporal prepositions. Text mining is a
specific type of data mining that focuses on unstructured text. Mining
association rules [3] is often used for extracting heuristic rules. Automatic
entity classification by ML models, in particular, decision trees, is another
rule mining technique [84].

###  5.2 Knowledge graphs

Machine learning has focused on extracting latent representations from
euclidean data including images, text and videos. The need to process graph
data has also important. Graph structures are natural representations in
domains, such as e-commerce [225, 224], drug discovery [41] and chemistry
[47], production and manufacturing, supply chain management [120] and network
optimization [177]. Graphs are a natural means for explanation of opaque
machine learning models. Therefore, they are used as input to machine learning
and output from machine learning, providing an important perspective for how
machine learning can support conceptual modeling. A knowledge graph $``$(i)
describes real world entities and their interrelations, organized in a graph,
(ii) defines possible classes and relationships of entities in a schema, (iii)
allows for potentially interrelating arbitrary entities with each other; and
(iv) covers various topical domains.$"$ [162].

Adding typed links between data from different sources is an active research
topic on semantic technologies [23]. Google, for example, extended semi-
automatic annotation procedures by automatic extraction of knowledge graphs
[189] based on existing sources, such as dbpedia [10], YAGO [199] or WordNet
[142].

Figure 15: Link prediction with TransR. Adapted from [124].

Knowledge graphs extract named connections between instances, called link
prediction between entities [124], which provide initial support for partial
conceptual models. With large datasets, the quality of triples found by
knowledge graphs is often low; that is, many links are tautological, or even
meaningless. Entity resolution, collective classification, and link prediction
can be used to construct consistent knowledge graphs based on probabilistic
soft logic [170]. A standard approach is to transform data into vector spaces
resembling principal component analysis (PCA). For instance, TransR proposed
by Lin et al. [124] projects entities $h$ (head) and$t$ (tail) from an entity
space into a r-relation space by a mapping function $M_{r}$ that supports
$h_{r}+r\approx t_{r}$ and, thus, finds a set of entities $t_{r}$ that fulfil
a triple $<h,r,t>$ (cf. Figure 15). $M_{r}$ is a vector embedding model that
is trained on a loss function that minimizes a distance $d(h+r,t)$ of ground
truth and $d\left(h+r,t\right)$ estimations. ML models based on embeddings are
generative models (cf. section 3.1.2) that encode entities and relationships
in vector spaces, make predictions into the input space, called decoding, and
measure the reconstruction error as an indicator for model performance. Hence,
graph embeddings are powerful models used for various graph analytical tasks.

Graph analytics use social network theory by analyzing distances and directed
connections (ties) of knowledge graphs to support semantic annotations, such
as similarity, centrality, community, and paths [214]. Similar to data
engineering, knowledge graphs are explored and missing elements predicted
(e.g., link prediction, completion and correction) [114]. When using knowledge
graphs to support machine learning, graphs are embedded into multidimensional
vector spaces by preserving a proximity measure defined on knowledge graph,$G$
[79]. Graph embedding (GE) reduces the dimensionality of a graph. Currently,
autoencoder models are used for embedding graph nodes and preserving non-
linear dependencies [211]. Graph embeddings are used for link prediction and
node classification and, thus, can be indirectly used for concept
classification, identification of relationships, and ontology learning [136].
There is a growing interest in graph neural networks, which consider graphs as
input, instead of tabular data. Graph neural networks (GNN) are extensions of
graph embedding with emphasis on deep learning architectures, such as
recurrent neural networks, convolutional neural networks and autoencoders
[222].

Graph embeddings (GE) and graph neural networks (GNN) require graph input that
meets quality requirements. Applications based on GE and GNN need to satisfy
all requirement types, in the ML development cycle. This makes conceptual
modeling methods and tools important assets for the development of GE/GNN-
based information systems. Research on knowledge graphs provide important
technologies for research on linked data and ontologies in general [93] and
conceptual modeling in particular, including reasoning and querying over
contextual data, and rule and axiom mining.

##  6 Summary and roadmap

In this paper, we align conceptual modeling and machine learning in both
directions. Due to the early stages of research related to this pairing,
challenges remain. Conceptual modeling was motivated by research on relational
databases [43] and procedural programming whereas machine learning is a child
of statistics and linear algebra. Although it is clear that conceptual
modeling can support data management of machine learning, many challenges
remain for supporting the development of model architectures, model training,
model testing, model optimization, deployment, and maintenance in information
systems. For example, deep convolutional neural network architectures consist
of multiple layers with different sizes taking different roles, such as
convolution and pooling layers, and application of different convolutional
kernels [115].

Many technical research issues emerge including:

  * •

Use of data ontologies and design patterns for data engineering

  * •

Alignment of data engineering with databases and big data stores

  * •

Models for mining data streams

  * •

Design patterns for model architectures

  * •

Process models for model development

  * •

Performance models for model development.

Because ML models are central to services delivered by information systems,
they also require alignment with enterprise architectures. Research issues
include the following:

  * •

Frameworks for aligning model architectures and service architectures and
enterprise architectures

  * •

Frameworks for alignment of performance metrics and key performance
indicators.

Decision making, which relies on machine learning must consider:

  * •

The quality and performance models for data-driven decision making (cf. [168])

  * •

Conceptual modeling in real-time data-driven decision making with batch and
streaming data.

Conceptual modeling is well positioned when it comes to structuring
requirements for complex systems, including ML-based systems. Recent proposals
structure data science development by views and goal models [127, 150]. In the
direction of machine learning for conceptual modeling, research issues are in
their infancies. Knowledge graphs extracted from data are promising areas with
clear connections to conceptual modeling. Merging knowledge graphs with formal
ontologies is a challenging, but also promising, research topic.

##  7 Conclusion

The fields of machine learning and conceptual modeling have been active areas
of research for a long time, making it reasonable to expect that it might be
advantageous to explore how one might complement the other. This paper has
identified possible synergies between the machine learning and conceptual
modeling and proposed a framework for conceptual modeling for machine
learning. Conceptual modeling can be helpful in supporting the design and
development phases of a machine learning-based information system. Feature
sets, especially, must be consistent so that data scientists can create valid
solutions. This can be best accomplished by including domain knowledge, as
represented by conceptual models. Inversely, machine learning techniques are
very successful at obtaining or scraping large amounts of data, which can be
used to identify concepts and patterns that could be useful for inclusion in
conceptual models. There are, of course, many challenges related to incorrect,
incomplete, or inconsistent knowledge. Nevertheless, it is feasible to pair
conceptual modeling with machine learning. This paper has identified some of
the challenges inherent in achieving this pairing in an attempt to lay the
groundwork for future research on combining conceptual modeling and machine
learning.

Acknowledgements

This paper was based on a keynote presentation given by the first author at
the International Conference on Conceptual Modeling. The authors wish to thank
Peter Chen and Carson Woo, and Oscar Pastor for their support of this paper,
Iaroslav Shcherbatyi for sharing his technical expertise in machine learning
and Michael Schrefl for identifying this topic. We also thank the anonymous
reviewers and Roman Lukyanenko for provided valuable insights and comments.

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N,\quad\operatorname{diam}(\mathrm{Bas}(\mathbf{P}_{N},x_{i}))\leq
c_{2}N^{-\frac{1}{d}}.$

  4. (d)

_All atoms are far from each other:_ there exists a constant $c_{2}$ such that

$\forall N\in\mathbb{N}^{\ast},\forall(1\leq j\neq k\leq N),\quad
d(x_{j},x_{k})\geq c_{1}N^{-\frac{1}{d}}.$

Notice that 5.1 follows from c, since the basins form a covering of
$\operatorname{spt}\mu$ by Corollary 4.3, while 5.2 is merely a rephrasing of
d.

##### Proof of a and b.

First note that by Corollary 4.3, the basins form a partition of $\nu$, thus

$\sum_{j=1}^{N}\nu(\mathrm{Bas}(\mathbf{P}_{N},x_{j}))=\lVert\nu\rVert,$ (5.3)

thus there exists an index $j\in\\{1,\ldots,N\\}$ for which

$\nu(\mathrm{Bas}(\mathbf{P}_{N},x_{j}))\leq\frac{\lVert\nu\rVert}{N},$

and by Lemma 4.4, this implies that

$\operatorname{diam}(\mathrm{Bas}(\mathbf{P}_{N},x_{j})\leq(\lVert\nu\rVert(c_{\mathrm{Bas}}N)^{-1})^{\frac{1}{d}}=\tilde{c}_{2}N^{-\frac{1}{d}}\quad\text{where}\quad\tilde{c}_{2}\coloneqq(\lVert\nu\rVert
c_{\mathrm{Bas}}^{-1})^{-\frac{1}{d}}.$

Similarly, there exists an index $i\in\\{1,\ldots,N\\}$ such that

$C_{A}\operatorname{diam}(\mathrm{Bas}(\mathbf{P}_{N},x_{i})\geq\nu(\mathrm{Bas}(\mathbf{P}_{N},x_{i}))\geq\frac{\lVert\nu\rVert}{N},$

which implies that

$\operatorname{diam}(\mathrm{Bas}(\mathbf{P}_{N},x_{i})\geq\tilde{c}_{2}N^{-\frac{1}{d}}\quad\text{where}\quad\tilde{c}_{2}\coloneqq(\lVert\nu\rVert
C_{A}^{-1})^{-\frac{1}{d}}.$

##### Proof of c.

Applying a we find that there exists $j\leq N$ such that

$\operatorname{diam}(\mathrm{Bas}(\mathbf{P}_{N},x_{j}))\leq\tilde{c}_{2}N^{-\frac{1}{d}}.$
(5.4)

Suppose that for some $t>1$ there exists $i\leq N$ such that

$\operatorname{diam}(\mathrm{Bas}(\mathbf{P}_{N},x_{i}))\geq
t\tilde{c}_{2}N^{-\frac{1}{d}}.$

We are going to show a contradiction when $t$ is too large (not depending on
$N$). For this, let us build a better competitor than $\mu_{N}$. We shall add
an extra point $q$ of $\mathrm{Bas}(\mathbf{P}_{N},x_{i})$ to irrigate a
“costly” ball around $q$, then remove the point $x_{j}$ and irrigate the
former basin $\mathrm{Bas}(\mathbf{P}_{N},x_{j})$ from a neighbouring basin
$\mathrm{Bas}(\mathbf{P}_{N},x_{k})$.

Let us find a “costly” ball. Consider

$q\in\operatorname{arg\,max}_{\mathrm{Bas}(\mathbf{P}_{N},x_{i})}z_{\mathbf{P}_{N}},$

which exists because $z_{\mathbf{P}_{N}}$ is Hölder-continuous thanks to 4.1
and basins are compact sets thanks to Corollary 4.3. We consider the ball
$B_{\varepsilon t\tilde{c}_{2}N^{-\frac{1}{d}}}(q)$ for some small
$\varepsilon\in(0,1)$ to be fixed later.

Now, we want to remove the point $x_{i}$ from the quantizer $\mu_{N}$ and to
irrigate the basin $\mathrm{Bas}(\mathbf{P}_{N},x_{i})$ from another basin
that is not too far, in order to control the extra cost. By 5.4 and Ahlfors-
regularity of $\nu$, for $s>1$ we have

$\nu(B_{s\tilde{c}_{2}N^{-\frac{1}{d}}}(x_{j})\setminus\mathrm{Bas}(\mathbf{P}_{N},x_{j}))\geq
c_{A}(s\tilde{c}_{2})^{d}N^{-1}-C_{A}\tilde{c}_{2}^{d}N^{-1}=(c_{A}s^{d}-C_{A})\tilde{c}_{2}^{d}N^{-1}.$

This is strictly positive if we take for example
$s\coloneqq(2C_{A}/c_{A})^{\frac{1}{d}}$, in which case there exists a point
$p$ such that

$p\in\mathrm{Bas}(\mathbf{P}_{N},x_{k})\cap
B_{s\tilde{c}_{2}N^{-1/d}}(x_{j})\setminus\mathrm{Bas}(\mathbf{P}_{N},x_{j})\quad\text{for
some}\quad k\neq j,$

because the basins form of covering of $\operatorname{spt}\nu.$

We are now ready to build our competitor $\mathbf{P}_{N}^{*}$, modifying
$\mathbf{P}_{N}$ according to the following sketch; the addition of curves
(which increase the $\alpha$-mass) are labeled by (A), while the removal of
curves (which decreases the $\alpha$-mass) are labeled by (R).

  1. (R1)

Remove all curves starting at $x_{j}$.

  2. (R2)

Remove all curves ending in $B_{\varepsilon\tilde{c}_{2}N^{-1/d}}(q)$.

  3. (A1)

Re-irrigate $\mathrm{Bas}(\mathbf{P}_{N},x_{j})$ by

     * •

bringing a mass $m_{j}$ from $x_{k}$ to $p$ following a
$\mathbf{P}_{N}^{x_{k}}$-good curve $\gamma$,

     * •

then concatenate an optimal traffic plan
$\mathbf{Q}^{1}\in\mathbf{TP}(m_{j}\delta_{p},\nu\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathrm{Bas}(\mathbf{P}_{N},x_{j}))$.

  4. (A2)

Add an optimal traffic plan $\mathbf{Q}^{2}$ from $m\delta_{q}$ to
$\nu\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(B_{\varepsilon
t\tilde{c}_{2}N^{-\frac{1}{d}}}(q)\setminus\mathrm{Bas}(\mathbf{P}_{N},x_{j}))$,
where $m$ is the mass of the latter and $\varepsilon>0$ is a small number to
be chosen later (independently from $N$ and $t$).

We start by doing the modifications along the existing network, corresponding
to (R1), (R2) and the first part of (A1). Setting
$\Gamma_{x_{j}}\coloneqq\\{\gamma:\gamma(0)=x_{j}\\}$ and
$\Gamma_{q}\coloneqq\\{\gamma:\gamma(+\infty)\in
B_{\varepsilon\tilde{c}_{2}N^{-\frac{1}{d}}}(q)\\}$, we define

$\mathbf{P}_{N}^{\prime}\coloneqq\mathbf{P}_{N}-\mathbf{P}_{N}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Gamma_{q}-\mathbf{P}_{N}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\Gamma_{x_{j}}\setminus\Gamma_{q})+m_{j}\delta_{\gamma}.$

Secondly, we add the new curves and pieces of curves corresponding to the
second part of (A1) and (A2). We set
$\nu^{\prime}\coloneqq(e_{\infty})_{\sharp}\mathbf{P}_{N}^{\prime}$, and
$\tilde{\mathbf{Q}}_{1}\coloneqq\mathbf{Q}_{1}+\iota_{\sharp}(\nu^{\prime}-m_{j}\delta
p)$ where $\iota:\mathbb{R}^{d}\to\Gamma^{d}$ denotes the canonical injection
which sends a point $x$ to the constant curve $\gamma_{x}\equiv x$. We define
our competitor $\mathbf{P}_{N}^{*}$ by

$\mathbf{P}_{N}^{*}\coloneqq\mathbf{P}_{N}^{\prime\prime}+\mathbf{Q}_{2}\quad\text{where}\quad\mathbf{P}_{N}^{\prime\prime}\in\mathbf{P}_{N}^{\prime}:\tilde{\mathbf{Q}}_{1}.$

We estimate the gain and cost of these operations. First of all, we have

$\displaystyle\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{*})$
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime\prime})+\mathbb{M}^{\alpha}(\mathbf{Q}_{2})\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime})+\mathbb{M}^{\alpha}(\mathbf{Q}_{2})+\mathbb{M}^{\alpha}(\mathbf{Q}_{1})$
(5.5)
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime})+C_{\mathsf{BOT}}C_{A}^{\alpha}(\varepsilon
t\tilde{c}_{2}N^{-1/d})^{1+d\alpha}+C_{\mathsf{BOT}}C_{A}^{\alpha}(\tilde{c}_{2}N^{-1/d})^{1+d\alpha}$
$\displaystyle=\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime})+C_{1}N^{-\left(\alpha+\frac{1}{d}\right)}(1+(\varepsilon
t)^{1+d\alpha}),$

for some constant $C_{1}$ which does dot depend on $N$. The $\alpha$-mass of
$\mathbf{P}_{N}^{\prime}$ may then be estimated through the first variation
formula Proposition 4.2 C:

$\displaystyle\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime})$
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})-\int_{B_{\varepsilon
t\tilde{c}_{2}N^{-1/d}}(q)}z_{\mathbf{P}_{N}}\mathop{}\mathopen{}\mathrm{d}\nu-\int_{\mathbb{R}^{d}}z_{\mathbf{P}_{N}}\mathop{}\mathopen{}\mathrm{d}(e_{\infty})_{\sharp}(\mathbf{P}_{N}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\Gamma_{x_{j}}\setminus\Gamma_{q}))+m_{j}z_{\mathbf{P}_{N}}(p)$
(5.6)
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})-\int_{B_{\varepsilon
t\tilde{c}_{2}N^{-1/d}}(q)}z_{\mathbf{P}_{N}}\mathop{}\mathopen{}\mathrm{d}\nu+m_{j}z_{\mathbf{P}_{N}}(p).$

Let us estimate $z_{\mathbf{P}_{N}}(p)$ from above and $z_{\mathbf{P}_{N}}$
from below on $B_{\varepsilon t\tilde{c}_{2}N^{-1/d}}(q)$. By Lemma 4.4, we
know that

$z_{\mathbf{P}N}(q)\geq
c_{H}\operatorname{diam}(\mathrm{Bas}(\mathbf{P}_{N},x_{i}))^{\beta}\geq
c_{H}(t\tilde{c}_{2})^{\beta}N^{-\beta/d},$

thus for every $y\in B_{\varepsilon t\tilde{c}_{2}N^{-1/d}}(q)$

$\displaystyle z_{\mathbf{P}_{N}}(y)$ $\displaystyle\geq
c_{H}(t\tilde{c}_{2})^{\beta}N^{-\beta/d}-C_{H}\lvert y-q\rvert^{\beta}$ (5.7)
$\displaystyle\geq(c_{H}-\varepsilon^{\beta}C_{H})(t\tilde{c}_{2})^{\beta}N^{-\beta/d}\geq(c_{H}/2)(t\tilde{c}_{2})^{\beta}N^{-\beta/d},$

provided we have chosen $\varepsilon^{\beta}\leq(c_{H}/2C_{H})$. Besides, by
Lemma 4.4 again

$\displaystyle z_{\mathbf{P}_{N}}(p)$
$\displaystyle\leq\sup_{y\in\mathrm{Bas}(\mathbf{P}_{N},x_{j})}\lvert
z_{\mathbf{P}_{N}}(y)-z_{\mathbf{P}_{N}}(p)\rvert+\sup_{y\in\mathrm{Bas}(\mathbf{P}_{N},x_{j})}z_{\mathbf{P}_{N}}(y)$
(5.8) $\displaystyle\leq
C_{H}((s+1)\tilde{c}_{2}N^{-1/d})^{\beta}+C_{H}^{\prime}(\tilde{c}_{2}N^{-1/d})^{\beta}$
$\displaystyle=CN^{-\beta/d}$

for some $C$ which does not depend on $N$. Reporting 5.7 and 5.8 in 5.6 yields

$\displaystyle\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime})$
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})-(c_{H}/2)t^{\beta}N^{-\beta/d}c_{A}(\varepsilon
t\tilde{c}_{2}N^{-1/d})^{d}+CN^{-\beta/d}C_{A}(s\tilde{c}_{2}N^{-1/d})^{d}$
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})-N^{-\left(\alpha+\frac{1}{d}\right)}(C_{2}\varepsilon^{d}t^{1+d\alpha}-C_{3}),$

for some constant $C_{2},C_{3}>0$ which do not depend on $N$.

Injecting this into 5.5, we get

$\displaystyle\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{*})\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})+N^{-\left(\alpha+\frac{1}{d}\right)}(C_{1}+C_{3}+C_{1}(\varepsilon
t)^{1+d\alpha}-C_{2}\varepsilon^{d}t^{1+d\alpha}).$

Notice that because $1+d\alpha-d=\beta>0$, it is possible to choose
$\varepsilon$ small enough, independently from $N$ and $t$ (e.g.555Recall that
we had the condition $\varepsilon^{\beta}\leq(c_{H}/2C_{H})$ for 5.7.
$\varepsilon^{\beta}=(C_{2}/2C_{1})\wedge(c_{H}/2C_{H})$) so that

$\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{*})\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})+N^{-\left(\alpha+\frac{1}{d}\right)}(C_{1}+C_{3}-(C_{2}/2)\varepsilon^{d}t^{1+d\alpha}).$

Now, if $t$ is too large, depending on the constants
$C_{1},C_{2},C_{3},\varepsilon$ which we stress do not depend on $N$, it leads
to
$\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{*})<\mathbb{M}^{\alpha}(\mathbf{P}_{N})$,
which contradicts the optimality of $\mu_{N}$, because by construction the
target measure of $\mathbf{P}_{N}^{*}$ is $\nu$, and its source measure is an
$N$-point quantizer. As a conclusion, c holds true.

##### Proof of d.

Take $t>0$ and suppose that there are two atoms $x_{j},x_{k}$ of $\mu_{N}$,
with $j\neq k$, of $\mu_{N}$ such that $d(x_{j},x_{k})\leq tN^{-1/d}$. We are
going to show a lower bound on $t>0$ (not depending on $N$). Choose a $i\leq
N$ such that

${\mathscr{L}}^{d}(\mathrm{Bas}(\mathbf{P}_{N},x_{i}))\geq\frac{\lVert\nu\rVert}{N}\quad\text{and}\quad\operatorname{diam}(\mathrm{Bas}(\mathbf{P}_{N},x_{i}))\geq\tilde{c}_{1}N^{-1/d}$

which is possible thanks to b. Up to interchanging $j$ with $k$, we may assume
that $i\neq j$ ($k$ may be equal to $i$, but it will not matter). The strategy
is very similar as what we did above: we shall add an extra point
$q\in\mathrm{Bas}(\mathbf{P}_{N},x_{i})$ to irrigate a “costly” ball around
$q$, while removing the point $x_{j}$ from the quantizer and irrigating the
basin $\mathrm{Bas}(\mathbf{P}_{N},x_{j})$ from the close point $x_{k}$.

Consider

$q\in\operatorname{arg\,max}_{\mathrm{Bas}(\mathbf{P}_{N},x_{i})}z_{\mathbf{P}_{N}},$

which is possibly because $z_{\mathbf{P}_{N}}$ is Hölder-continuous thanks to
4.1 and basins are compact sets thanks to Corollary 4.3. Take some small
$\varepsilon\in(0,1)$ that we shall fix later. We build a better competitor
thanks to the following construction.

  1. (R1)

Remove all curves ending in $B_{\varepsilon\tilde{c}_{1}N^{-1/d}}(q)$, of
total mass $m$.

  2. (A1)

Add an optimal traffic plan
$\mathbf{Q}^{1}\in\mathbf{TP}(m\delta_{q},\nu\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}B_{\varepsilon\tilde{c}_{1}N^{-1/d}}(q))$
to irrigate $\nu\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}B_{\varepsilon\tilde{c}_{1}N^{-1/d}}(q)$
again.

  3. (A2)

Irrigate $\mathrm{Bas}(\mathbf{P}_{N},x_{j})$ from the point $x_{k}$ instead
of $x_{j}$ by concatenating a (unit-speed parameterization of) the segment
$[x_{k},x_{j}]$ to all curves starting at $x_{j}$.

The removal (R1) produces the new traffic plan

$\mathbf{P}^{\prime}_{N}\coloneqq\mathbf{P}_{N}-\mathbf{P}_{N}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Gamma_{q}\quad\text{where}\quad\Gamma_{q}\coloneqq\\{\gamma:\gamma(\infty)\in
B_{\varepsilon\tilde{c}_{1}N^{-1/d}}(q)\\}.$

We show as above by Lemma 4.4 that for every $y\in
B_{\varepsilon\tilde{c}_{1}N^{-1/d}}(q)$

$z_{\mathbf{P}_{N}}(y)\geq(c_{H}/2)(t\tilde{c}_{1})^{\beta}N^{-\beta/d},$
(5.9)

provided we have chosen $\varepsilon^{\beta}\leq(c_{H}/2C_{H})$. Thus the cost
gain can be estimated by using the first variation formula Proposition 4.2 C:

$\displaystyle\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime})$
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})-\alpha\int_{B_{\varepsilon\tilde{c}_{1}N^{-1/d}}(q)}z_{\mathbf{P}_{N}}\mathop{}\mathopen{}\mathrm{d}\nu$
(5.10)
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})-C_{A}(\varepsilon\tilde{c}_{1}N^{-1/d})^{d}(c_{H}/2)N^{-\beta/d}$
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})-C_{1}\varepsilon^{d}N^{-\left(\alpha+\frac{1}{d}\right)}.$

For the addition of the (pieces of) curves (A1) and (A2), we denote by
$\gamma_{k,j}$ the unit-speed parameterized segment from $x_{k}$ to $x_{j}$,
$\mathbf{P}_{N,j}^{\prime}\coloneqq\mathbf{P}_{N}^{\prime}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\\{\gamma:\gamma(0)=x_{j}\\}$,
$m_{j}^{\prime}\coloneqq\lVert\mathbf{P}_{N,j}^{\prime}\rVert$ and we set
$\mathbf{Q}_{2}\coloneqq m^{\prime}_{j}\delta_{\gamma_{k,j}}+(\iota\circ
e_{0})_{\sharp}(\mathbf{P}_{N}^{\prime}-\mathbf{P}_{N,j}^{\prime})$. We define
our competitor $\mathbf{P}_{N}^{*}$ by

$\mathbf{P}_{N}^{*}\coloneqq\mathbf{P}_{N}^{\prime\prime}+\mathbf{Q}_{1}\quad\text{where}\quad\mathbf{P}_{N}^{\prime\prime}\in\mathbf{Q}_{2}:\mathbf{P}_{N}^{\prime}.$

From c, we know that

$m_{j}^{\prime}\leq m_{j}\leq
C_{A}(\operatorname{diam}(\mathrm{Bas}(\mathbf{P}_{N},x_{j})))^{d}\leq
C_{A}c_{2}^{d}N^{-1}$

and we compute, using 5.10:

$\displaystyle\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{*})$
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime\prime})+\mathbb{M}^{\alpha}(\mathbf{Q}_{1})\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime})+\mathbb{M}^{\alpha}(\mathbf{Q}_{1})+\mathbb{M}^{\alpha}(\mathbf{Q}_{2})$
(5.11)
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime})+C_{\mathsf{BOT}}C_{A}^{\alpha}(\varepsilon\tilde{c}_{1}N^{-1/d})^{1+d\alpha}+(m_{j}^{\prime})^{\alpha}(tN^{-1/d})$
$\displaystyle=\mathbb{M}^{\alpha}(\mathbf{P}_{N})+N^{-\left(\alpha+\frac{1}{d}\right)}(-C_{1}\varepsilon^{d}+C_{2}\varepsilon^{1+d\alpha}+t).$

Taking $\varepsilon>0$ such that $\varepsilon^{\beta}\leq c_{H}/2C_{H}$ and
$\varepsilon^{\beta}\leq C_{1}/2C_{2}$ (e.g. take $\varepsilon^{\beta}$ to be
the minimum of the twi), we get
$C_{2}\varepsilon^{1+d\alpha}\leq(C_{1})/2\varepsilon^{d}$ and thus

$\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{*})\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})+N^{-\left(\alpha+\frac{1}{d}\right)}(t-(C_{1}/2)\varepsilon^{d}),$

which leads to a contradiction if $t$ is too small (independently from $N$).
Hence $t$ is lower bounded by some constant $c_{1}>0$ and d holds true. ∎

###  5.2 Inner and outer ball property of the basins

######  Proposition 5.2.

Let $\nu\in\mathscr{M}^{+}(K)$ be a $d$-Ahlfors regular measure with constants
$0<c_{A}\leq C_{A}$. Let $\mu_{N}=\sum_{i=1}^{N}m_{i}\delta_{x_{i}}$ be a
$N$-point optimal quantizer of $\nu$ and $\mathbf{P}_{N}$ an optimal traffic
plan in $\mathbf{TP}(\mu_{N},\nu)$. There are constants $c,C>0$ depending only
on $\alpha,d,c_{A},C_{A}$ such that for all $i\in\\{1,\ldots,N\\}$,

$\mathrm{Bas}(\mathbf{P}_{N},x_{i})\subseteq B(x_{i},CN^{-1/d}),$ (5.12)

and

$B(x_{i},cN^{-1/d})\subseteq\mathbb{R}^{d}\setminus\bigcup_{j\neq
i}\mathrm{Bas}(\mathbf{P}_{N},x_{j}).$ (5.13)

######  Remark 5.3.

In particular, if $\nu={\mathscr{L}}^{d}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Omega$ for some open bounded set
$\Omega$ with Lipschitz boundary, then for every source $x_{i}$ such that
$d(x_{i},\partial\Omega)>cN^{-1/d}$, 5.12 and 5.13 rewrite as

$B(x_{i},cN^{-1/d})\subseteq\mathrm{Bas}(\mathbf{P}_{N},x_{i})\subseteq
B(x_{i},CN^{-1/d}).$

Besides, this translates as uniform inner and outer ball properties of optimal
partitions (solutions to 2.24). Indeed, by the equivalence between the optimal
quantization and optimal partition problems stated in 2.7, solutions
$(\Omega_{i})_{1\leq i\leq N}$ to 2.24 are actually
${\mathscr{L}}^{d}$-equivalent to basins
$(\mathrm{Bas}(\mathbf{P}_{N},x_{i}))_{1\leq i\leq N}$ for some traffic plan
$\mathbf{P}_{N}$ with sources $(x_{i})_{1\leq i\leq N}$.

Finally, we remark that the number of points such that
$d(x_{i},\partial\Omega)>cN^{-1/d}$ is $N+O(N^{1-1/d})$ because the other
points convey a mass $\approx N^{-1}$ by Proposition 5.4, and their basins are
included in $\\{d(\cdot,\partial\Omega)<C^{\prime}N^{-1/d}\\}$ for some
$C^{\prime}$, a set of volume $\approx N^{-1/d}$. However, without extra
assumptions on $\Omega$, some points $x_{i}$ may very well belong to
$\mathbb{R}^{d}\setminus\bar{\Omega}$. Thus is ruled out for example when
$\Omega$ is convex, but then it is not guaranteed that $x_{i}\in\Omega$ for
all $x_{i}$’s for general $\Omega$.

###### Proof of Proposition 5.2.

The outer ball property 5.12 holds with $C\coloneqq c_{2}$, the constant in c
in the proof of 5.1. For the inner ball property 5.13, assume that
$d(x_{i},\mathrm{Bas}(\mathbf{P}_{N},x_{j}))\leq\varepsilon N^{-1/d}$ for some
$i\neq j$. We shall find a lower bound on $\varepsilon$ that depends only on
$(\alpha,d,c_{A},C_{A},\operatorname{diam}(\nu))$. If $\varepsilon\leq
c_{1}/2$ where $c_{1}$ is the separation constant in 5.2, then taking a point
$x\in B(x_{i},\varepsilon N^{-1/d})\cap\mathrm{Bas}(\mathbf{P}_{N},x_{j})$, we
have

$d(x,x_{j})\geq d(x_{i},x_{j})-d(x_{i},x)\geq(c_{1}/2)N^{-1/d},$

thus taking $\gamma_{j}$ a $\mathbf{P}_{N}$-good curve from $x_{j}$ to $x$,
its length is greater than $(c_{1}/2)N^{-1/d}$ hence

$z_{\mathbf{P}_{N}}(x)=\int_{\gamma_{j}}\Theta_{\mathbf{P}_{N}^{x_{j}}}^{\alpha-1}\geq(c_{1}/2)N^{-1/d}\nu(\mathrm{Bas}(\mathbf{P}_{N},x_{j}))^{\alpha-1}\geq\frac{1}{2}{c_{1}(C_{A}c_{2})^{\alpha-1}}N^{-\beta/d}.$

As a consequence, setting
$c^{\prime}\coloneqq\frac{1}{2}{c_{1}(C_{A}c_{2})^{\alpha-1}}$ and
$c^{\prime\prime}\coloneqq(c^{\prime}/2C_{H})^{1/\beta}$, for every $y\in
B(x,c^{\prime\prime}N^{-1/d})\cap\operatorname{spt}\nu$, by 4.1

$z_{\mathbf{P}_{N}}(y)\geq z_{\mathbf{P}_{N}}(x)-C_{H}\lvert
y-x\rvert^{\beta}\geq c^{\prime}/2N^{-\beta/d}.$

If $\varepsilon\leq c^{\prime\prime}$ we build a competitor
$\mathbf{P}_{N}^{\prime}$ by removing from $\mathbf{P}_{N}$ all curves going
to $B(x,\varepsilon N^{-1/d})$, and adding an optimal traffic plan
$Q\in\mathbf{TP}(m_{\varepsilon}\delta_{x_{i}},\nu\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}B(x,\varepsilon N^{-1/d})$ where
$m_{\varepsilon}\coloneqq\nu(B(x,\varepsilon N^{-1/d}))$. Using the first
variation formula Proposition 4.2 C and the subadditivity of the
$\alpha$-mass, we get by optimality of $\mu_{N}$

$\displaystyle\mathbb{M}^{\alpha}(\mathbf{P}_{N})\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N}^{\prime})$
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})-\alpha\int_{B(x,\varepsilon
N^{-1/d})}z_{\mathbf{P}_{N}}\mathop{}\mathopen{}\mathrm{d}\nu+C_{\mathsf{BOT}}(2\varepsilon
N^{-1/d})(C_{A}\varepsilon^{d}N^{-1})^{\alpha}$
$\displaystyle\leq\mathbb{M}^{\alpha}(\mathbf{P}_{N})-(\alpha
c_{A}c^{\prime}/2)\varepsilon^{d}N^{-\left(\alpha+\frac{1}{d}\right)}+2C_{\mathsf{BOT}}C_{A}^{\alpha}\varepsilon^{1+d\alpha}N^{-\left(\alpha+\frac{1}{d}\right)}.$

We get a contradiction when $\varepsilon$ is smaller than some constant
$c^{\prime\prime\prime}>0$ (depending only on
$(\alpha,c_{a},c^{\prime},C_{\mathsf{BOT}},C_{A})$) because $d<1+d\alpha$.
During the reasoning we made, recall that we assumed $\varepsilon\leq c_{1}/2$
then $\varepsilon\leq c^{\prime\prime}$, thus we must have:

$\varepsilon\geq
c\coloneqq\min\\{c_{1}/2,c^{\prime\prime},c^{\prime\prime\prime}\\},$

and 5.13 holds with this constant $c>0$. ∎

###  5.3 Uniformity bounds for masses and energies

######  Proposition 5.4.

Let $\nu\in\mathscr{M}^{+}(K)$ be a $d$-Ahlfors regular measure with constants
$c_{A},C_{A}$. Let $\mu_{N}=\sum_{i=1}^{N}m_{i}\delta_{x_{i}}$ be an $N$-point
optimal quantizer (i.e. a solution to 2.17) and
$\mathbf{P}_{N}\in\mathbf{OTP}^{\alpha}(\mu_{N},\nu)$. There are constants
$c,C>0$ depending only on $(\alpha,d,c_{A},C_{A})$ such that for all
$i\in\\{1,\ldots,N\\}$,

$cN^{-1}\leq m_{i}\leq CN^{-1}$

and

$cN^{-(\alpha+1/d)}\leq\mathbf{d}^{\alpha}(m_{i}\delta_{x_{i}},\nu\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathrm{Bas}(\mathbf{P}_{N},x_{i}))\leq
CN^{-(\alpha+1/d)}.$

###### Proof.

The upper bounds come from the fact that, by 5.1, for every
$i\in\\{1,\ldots,N\\}$, $\mathrm{Bas}(\mathbf{P}_{N},x_{i})$ has diameter less
than $c_{2}N^{-1/d}$, thus

$m_{i}=\nu(\mathrm{Bas}(\mathbf{P}_{N},x_{i}))\leq C_{A}c_{2}^{d}N^{-1},$

and thus from the usual estimate of branched transport cost we get

$\mathbf{d}^{\alpha}(m_{i}\delta_{x_{i}},\nu\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathrm{Bas}(\mathbf{P}_{N},x_{i}))\leq
C_{\mathsf{BOT}}(c_{2}N^{-1/d})(C_{A}c_{2}^{d}N^{-1})^{\alpha}.$

The lower bounds results from Proposition 5.2, which implies that ($c,C$ being
the inner and outer ball constants):

$\mu\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathrm{Bas}(\mathbf{P}_{N},x_{i})\geq\mu\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}B(x_{i},cN^{-1/d}),$

and thus

$\mu(\mathrm{Bas}(\mathbf{P}_{N},x_{i}))\geq c_{A}c^{d}N^{-1}.$

Since $\mu$ is $d$-Ahlfors regular, it may be written
$\mu=f{\mathscr{L}}^{d}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}X$ with $c_{A}/\omega_{d}\leq
f\leq C_{A}/\omega_{d}$ on some Borel set $X\subseteq\operatorname{spt}\nu$.
Therefore, $C_{A}^{-1}\nu\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathrm{Bas}(\mathbf{P}_{N},x_{i})$
is a measure which is absolutely continuous with respect to Lebesgue, with
density in $[0,1]$, and total mass greater than $m\coloneqq
c_{A}/C_{A}c^{d}N^{-1}$, thus

$\mathbf{d}^{\alpha}(m_{i}\delta{x_{i}},\mu\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathrm{Bas}(\mathbf{P}_{N},x_{i}))\geq
C_{A}^{\alpha}e_{\alpha,d}m^{\alpha+\frac{1}{d}},$

where $e_{\alpha,d}>0$ is the constant from the optimal shape problem studied
in [PSX19], whose definition is given in 2.30. ∎

Acknowledgments. P.P. acknowledges the academic leave provided by Inria Paris
(team Mokaplan) for the year 2022-2023, and the “Young Researcher’s Project”
funding provided by the Scientific Council of Université Paris-Dauphine. M.P.
acknowledges support from the Chilean Fondecyt Regular grant 1210426 entitled
"Rigidity, stability and uniformity for large point configurations".

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