# Geometry of Data

Parvaneh Joharinad and Jürgen Jost

###### Abstract

Topological data analysis asks when balls in a metric space $(X,d)$ intersect.
Geometric data analysis asks how much balls have to be enlarged to intersect.
We connect this principle to the traditional core geometric concept of
curvature. This enables us, on one hand, to reconceptualize curvature and link
it to the geometric notion of hyperconvexity. On the other hand, we can then
also understand methods of topological data analysis from a geometric
perspective.

##  1 Introduction

Many data sets come with a basic geometric structure, distances between data
points. It is therefore natural to use geometric methods to analyze such data.
The deepest geometric concepts, however, were developed in the 19th century
for smooth manifolds, more precisely Riemannian manifolds. And the most
fundamental concept there is curvature. In the 20th century, notions of
curvature were successfully generalized to more general classes of spaces.
Still, those spaces, like geodesic length spaces, are typically not discrete,
in contrast to data sets. Thus, we have found it desirable to rethink
fundamental geometric concepts from a more abstract perspective that also
naturally includes discrete spaces. Of course, there are ideas and approaches
that we can build upon, most importantly those pioneered by Gromov [21, 22].
From such a perspective, the distinction between discrete and connected spaces
is partly one of scale. From a large scale perspective, spaces from those two
classes may look alike.

Such a large scale perspective is still quantitative, hence geometric, and is
therefore different from a qualitative topological approach. Nevertheless, as
we shall see, there are important links between the two. In particular, we can
look at the successful topological data analysis method of persistent homology
from a geometric perspective.

Topological data analysis asks when balls in a metric space $(X,d)$ intersect.
This is a qualitative concept, but the data analysis method of persistent
homology makes this quantitative through the dependence on the radii of the
balls. Geometric data analysis, as we conceive it in this contribution, asks
how much balls have to be enlarged to intersect. And as we shall see, this is
captured by a suitable concept of curvature. And curvature, from a general
perspective as adopted here, quantifies convexity. Therefore, convexity and
its strengthening as hyperconvexity will be our basic concepts.

##  2 Preliminaries from metric geometry

Let $(X,d)$ be a metric space. $x,y,\dots$ will be points in $X$, and they
thus have a distance $d(x,y)$. A continuous path $c:[0,1]\longrightarrow X$
with $x=c(0),y=c(1)$ has length

$\mathit{l}(c):=\sup\sum_{i=1}^{i=n}d(c(t_{i}),c(t_{i-1})).$

The supremum here is taken over all partitions of $[0,1]$, with
$t_{0}=0,t_{n}=1$. $(X,d)$ is called a _length space_ if for all $x,y$,

$d(x,y)=\inf\\{\mathit{l}(c):\>\text{c is a path
between}\;x\;\text{and}\;y\\}.$

A length space $(X,d)$ is called geodesic if this infimum is always realized,
that is, any $x,y\in X$ can be connected by a shortest path
$c:[0,1]\longrightarrow X$, i.e.

$d(x,y)=\mathit{l}(c).$

Thus, the distance between $x$ and $y$ is realized by some curve, a shortest
geodesic.
Every complete locally compact length space is a geodesic space. However,
there is another way to determine whether a complete metric space is a
geodesic (resp. length) space by checking the existence of mid-points (resp.
approximate midpoints).

######  Definition 2.1.

$m\in X$ is a midpoint between $x,y$ if

$d(x,m)=d(m,y)=\frac{1}{2}d(x,y).$

We may also say that a pair of points $x,y\in X$ has approximate midpoints if
for every $\epsilon>0$ there exists $m_{\epsilon}\in X$ with

$\max\\{d(m_{\epsilon},x),d(m_{\epsilon},y)\\}\leq\frac{1}{2}d(x,y)+\epsilon$

We observe

######  Lemma 2.1.

Every pair of points in a geodesic space (resp. length space) has at least one
midpoint (resp. approximate midpoints).
The inverse is true provided that the metric space is complete. ∎

In the sequel,

$B(x,r):=\\{y\in X:d(x,y)\leq r\\}$

will always be the closed ball centered at $x$ with radius $r\geq 0$.

######  Definition 2.2.

$(X,d)$ is totally convex if for any $x_{1},x_{2}\in X,r_{1},r_{2}>0$ with

${r_{1}+r_{2}\geq d(x_{1},x_{2})},$

we have

$B(x_{1},r_{1})\cap B(x_{2},r_{2})\neq\emptyset.$

Any radii $r_{i}$ will be $>0$ in the sequel.
Again, an easy lemma

######  Lemma 2.2.

Geodesic spaces are totally convex.∎

Length spaces are not necessarily totally convex, as they need not be
complete. An example is ${\mathbb{R}}^{2}\setminus{0}$ with the length
structure induced by the Euclidean distance.

Let us formulate Definition 2.2 as a

######  Principle 2.1.

Two balls that can intersect do intersect.

We shall now introduce a fundamental quantity. For

$r_{1}+r_{2}\geq d(x_{1},x_{2})$

we put

$\displaystyle\rho((x_{1},x_{2}),(r_{1},r_{2}))$ $\displaystyle:=$
$\displaystyle\inf_{x\in X}\max_{i=1,2}\frac{d(x_{i},x)}{r_{i}}$ (1)
$\displaystyle\rho(x_{1},x_{2})$ $\displaystyle:=$
$\displaystyle\sup_{r_{1},r_{2}}\rho((x_{1},x_{2}),(r_{1},r_{2}))$ (2)

If $\rho(x_{1},x_{2})=1$ for each pair of points $x_{1},x_{2}\in X$, then the
existence of approximate midpoints is guaranteed, and $X$ is a length space
provided that it is a complete metric space. If, moreover, the infimum is
attained for each pair by some $x_{0}\in X$, then $X$ is a geodesic space
provided that it is complete.
Another obvious

######  Lemma 2.3.

When $X$ is complete the supremum in (2) is realized by
$r_{1}=r_{2}=\frac{1}{2}d(x_{1},x_{2})$, that is

$\rho(x_{1},x_{2})=\inf_{x\in
X}\max_{i=1,2}\frac{2d(x_{i},x)}{d(x_{1},x_{2})}.$ (3)

Moreover, $\rho(x_{1},x_{2})=1$ is achieved for some $x$ when

$d(x_{1},x)+d(x_{2},x)=d(x_{1},x_{2}),$

that is, when $x$ is a midpoint of $x_{1},x_{2}$.

∎

Thus, we want to find points between two points $x_{1}$ and $x_{2}$, and
quantify to what extent that can fail.
Therefore, in the realm of complete metric spaces, the more (2) deviates from
$1$ the less is the chance to approximate distances by lengths of connecting
paths.

A key idea now is to extend this to three points.

##  3 Tripod spaces

######  Definition 3.1.

A geodesic length space $(X,d)$ is a tripod space if for any three points
$x_{1},x_{2},x_{3}\in X$, there exists a median, that is, a point $m\in X$
with

$d(x_{i},m)+d(x_{j},m)=d(x_{i},x_{j}),\>\text{for}\>1\leq i<j\leq 3.$

We note that for a median, we have

$d(x_{1},x_{2})+d(x_{2},x_{3})+d(x_{3},x_{1})=2(d(x_{1},m)+d(x_{2},m)+d(x_{3},m)).$
$x_{1}$$x_{2}$$x_{3}$$m$

Most metric spaces are not tripod spaces. For instance, Riemannian manifolds
of dimension $>1$ do not satisfy tripod property. Nevertheless, there are
examples that will be important for us:

  * •

Metric trees

  * •

$L^{\infty}$-spaces

  * •

and more generally, hyperconvex spaces (to be defined shortly)

If such a median exists it will be a minimizer for the sum of the distances to
the corresponding triple $x_{1},x_{2},x_{3}$. Such a point is called a Fermat
point.

Our strategy will then be to quantify the deviation from the tripod property.

We get the existence of tripods if the following more general condition is
satisfied. For any $x_{1},x_{2},x_{3}\in X$ which do not lie on a geodesic,
and $r_{i}+r_{j}\geq d(x_{i},x_{j})$, $1\leq i<j\leq 3$,

$\bigcap\limits_{i=1}^{3}B(x_{i},r_{i})\neq\emptyset.$

This leads to

######  Principle 3.1.

Three balls that can intersect do intersect.

To explore this principle, and the deviation from it, we shall now introduce a
3-point analogue of (1), (2) For $x_{1},x_{2},x_{3}\in X$ and $r_{i}+r_{j}\geq
d(x_{i},x_{j})$,

$\displaystyle\rho((x_{1},x_{2},x_{3}),(r_{1},r_{2},r_{3}))$ $\displaystyle:=$
$\displaystyle\inf_{x\in X}\max_{i=1,2,3}\frac{d(x_{i},x)}{r_{i}}$ (4)
$\displaystyle\rho(x_{1},x_{2},x_{3})$ $\displaystyle:=$
$\displaystyle\sup_{r_{i}+r_{j}\geq d(x_{i},x_{j}),i\neq
j}\rho((x_{1},x_{2},x_{3}),(r_{1},r_{2},r_{3})).$ (5)

This is uniquely solved by the Gromov products

$\displaystyle r_{1}$
$\displaystyle=\frac{1}{2}(d(x_{1},x_{2})+d(x_{1},x_{3})-d(x_{2},x_{3})),$
$\displaystyle r_{2}$
$\displaystyle=\frac{1}{2}(d(x_{1},x_{2})+d(x_{2},x_{3})-d(x_{1},x_{3})),$
$\displaystyle r_{3}$
$\displaystyle=\frac{1}{2}(d(x_{1},x_{3})+d(x_{2},x_{3})-d(x_{1},x_{2})).$ (6)

Remark: It is obvious that $\rho((x_{1},x_{2},x_{3}),(r_{1},r_{2},r_{3}))\geq
1$. Moreover, this quantity is bounded from above by $2$ if $X$ is complete.
If (with $r_{1},r_{2},r_{3}$ defined by (3)) $\rho(x_{1},x_{2},x_{3})=1$ and
the infimum is attained by some $m$, then we have a tripod construction or
equivalently a Fermat point. This implies that there exists an intermediate
point through which each pair $x_{i},x_{j}$ can be connected.

######  Definition 3.2.

An $m$ attaining the infimum in (4) is called a weighted circumcenter.

A weighted circumcenter solves an optimization problem in ${\mathbb{R}}^{3}$
with respect to the $\mathit{l}_{\infty}$ norm. The larger the value of
$\rho(x_{1},x_{2},x_{3})$ is, the less optimal the weighted circumcenter as
the interconnecting point will be.
We observe here

######  Lemma 3.1.

Weighted circumcenters exist and are unique for triangles in $CAT(0)$ spaces
(Alexandrov’s generalization of Riemannian manifolds of sectional curvature
$\leq 0$).

##  4 Hyperconvexity

We shall now extend the above principle to arbitrary numbers of points.

######  Definition 4.1.

$(X,d)$ is hyperconvex if for any family $\\{x_{i}\\}_{i\in I}\subset X$ and
$r_{i}+r_{j}\geq d(x_{i},x_{j})$ for $i,j\in I$,

$\bigcap\limits_{i\in I}B(x_{i},r_{i})\neq\emptyset$

In a totally convex metric space, $r_{i}+r_{j}\geq d(x_{i},x_{j})$ can be
replaced by $B(x_{i},r_{i})\cap B(x_{j},r_{j})\neq\emptyset$ for all $i,j\in
I$. Thus, when balls intersect pairwise, they also have a common intersection.
This leads to our final

######  Principle 4.1.

Balls that can intersect do intersect.

We observe

######  Lemma 4.1.

Hyperconvex spaces are tripod spaces. ∎

We list some important properties of hyperconvex spaces

######  Theorem 4.1.

  1. a)

Hyperconvex spaces are complete and contractible to each of their points [5].

  2. b)

$X$ is hyperconvex iff every $1-$Lipschitz map from a subspace of any metric
space $Y$ to $X$ can be extended to a $1-$Lipschitz map over $Y$ [5].

  3. c)

Every metric space is isometrically embedded in a hyperconvex space, called
its hyperconvex hull. The hyperconvex hull of a compact space is compact and
that of a finite space is a simplicial complex. [29, 16]

We now describe the isometric embedding in the part (c) and the construction
of the hyperconvex hull, in order to understand the specific choice of radii
in (1) and (4). By the Kuratowski embedding, every metric space $(X,d)$ is
isometrically embedded in the space of bounded functions on $X$ equipped with
the supremum norm, i.e. $l_{\infty}(X)$, via the map $x\mapsto d(x,.)$ which
we denote by $x\mapsto d_{x}$ for simplicity.
$l_{\infty}(X)$ contains the subspace $E(X)$ consisting of all functions $f$
that are minimal subject to the relation

$f(x)+f(y)\geq d(x,y),\;\forall x,y\in X.$ (7)

It has been shown in [29, 16, 35] that $E(X)$ is a hyperconvex space
containing the image of $X$ under the Kuratowski embedding isometrically, and
$E(X)$ is minimal in the sense that it is isometrically embeddable in any
other such hyperconvex space.
The radii in (1) and (4) are functions on a $2$-point space and a $3$-point
spaces respectively, satisfying (7).
If $X$ is a finite metric space with $|X|=n$, the space of all functions
satisfying (7) is a polyhedron in the finite vector space ${\mathbb{R}}^{n}$
obtained by the intersection of the closed half spaces $f_{i}+f_{j}\geq
d(x_{i},x_{j})$ for $1\leq i<j\leq n$. Therefore, the interior of every face
$S$ of this polyhedron is the intersection of some hyperplanes
$f_{i}+f_{j}=d(x_{i},x_{j})$. We can then define a graph $G(S)$ with vertex
set $X$, corresponding to the symmetric relation defined by that face. More
precisely, $x_{i}$ is connected to $x_{j}$ with an edge in $G(S)$ if for $f\in
S^{\circ}$ we have $f_{i}+f_{j}=d(x_{i},x_{j})$. Now, $E(X)$ is the union of
compact faces of this polyhedron and moreover the graph corresponding to each
such face is a spanning graph, that is, every vertex is connected to at least
one other vertex in this graph. This construction was first introduced in
[16], where a combinatorial dimension for finite metric spaces was defined as
the maximal dimension of a face in its hyperconvex hull. The hyperconvex hull
of finite metric spaces was studied further in [6, 17] and from a different
perspective in [14, 40] to obtain the metric fan of a finite set. In [40] a
software tool was presented to visualize these hyperconvex hulls. The problem
of finding faces of $E(X)$, when $X$ is finite, as a linear programming
problem was also studied in [28, 15]
In the special case $X=\\{x_{1},x_{2}\\}$ with distance
$d_{12}=d(x_{1},x_{2})$, the corresponding polyhedron is the half plane
$f_{1}+f_{2}\geq d_{12}$ cut by the coordinate planes $f_{i}=0,\;i=1,2$, which
has only one compact face, the line segment $f_{1}+f_{r}=d_{12}$ connecting
$(d_{12},0)$ to $(0,d_{12})$, i.e., $[d_{x_{2}},d_{x_{1}}]$. Every point in
this polyhedron can be reached through a ray passing this line segment and the
midpoint of this segment, that is $\frac{1}{2}(d_{12},d_{12})$ is the
corresponding radius function in (3). The space of all such radius functions
is illustrated in Figure 1(a).
Similarly, one can see that for $X=\\{x_{1},x_{2},x_{3}\\}$, using the same
notation $d_{ij},\;1\leq i,j\leq 3$ for pairwise distances, the corresponding
polyhedron is the intersection of the half-spaces

$f_{i}+f_{j}\geq d_{ij},\>1\leq i<j\leq 3$ (8)

and the coordinate half spaces $f_{i}\geq 0$ for $i=1,2,3$. Moreover, the
hyperconvex hull, colored in blue in Figure 1(b), is the union of three
segments each of which connect a distance function $d_{x_{i}}$ to the function
$r=(r_{1},r_{2},r_{3})$ defined in (3).

(a) 2-point

(b) 3-point

Figure 1: a) The yellow area is the set of all possible radius functions on
$2$ points and the line segment colored in blue refers to the minimal ones.
b) The three dimensional polyhedron is the set of all possible radius
functions on $3$ points and the tripod consisting of three line segments
colored in blue refers to the minimal ones.

For the analysis of discrete metric spaces, some variants of the notion of
hyperconvexity are well suited, c.f [33, 20, 23, 24].

######  Definition 4.2.

$(X,d)$ is $\delta$-hyperbolic ($\delta\geq 0$) if for any family
$\\{B(x_{i},r_{i})\\}_{i\in I}$ with $r_{i}+r_{j}\geq d(x_{i},x_{j})$,

$\bigcap_{i\in I}B(x_{i},\delta+r_{i})\neq\emptyset.$ (9)

######  Definition 4.3.

$(X,d)$ is $\lambda$-hyperconvex ($\lambda\geq 1$) if for every family
$\\{B(x_{i},r_{i})\\}_{i\in I}$ with $r_{i}+r_{j}\geq d(x_{i},x_{j})$,

$\bigcap\limits_{i\in I}B(x_{i},\lambda r_{i})\neq\emptyset.$ (10)

Of course, $0$-hyperbolicity and $1$-hyperconvexity are simply hyperconvexity.
For large radii, $\delta$ insignificant, and the concept of
$\delta$-hyperbolicity is therefore good for asymptotic considerations. In
contrast, $\lambda$-hperconvexity is invariant under scaling the metric $d$,
and it can therefore capture scaling invariant properties of a metric space.
The preceding concepts allow for a quantification of the deviation from
hyperconvexity. The following results are known.

######  Theorem 4.2.

Hilbert spaces are $\sqrt{2}$-hyperconvex. Reflexive and dual Banach spaces
are $2$-hyperconvex. Therefore, for a measure space $(X,\mu)$,
$L^{p}(X,\mu),1<p<\infty$, are $2$-hyperconvex, and if $X$ is finite,
$L^{1}(X,\mu)$ is also $2$-hyperconvex. $L^{\infty}(X,\mu)$ is hyperconvex.
[33, 20]

##  5 Relation with Topological Data Analysis (TDA)

######  Definition 5.1.

For a family $(x_{i})_{i\in I}$ in a metric space $(X,d)$ and $r>0$, we define
the _Čech complex_ $\check{C}_{r}((x_{i}),X)$ containing a $q$-simplex
whenever

$\bigcap_{i=1,\dots,q+1}B(x_{i},r)\neq\emptyset.$

Here $(x_{i})_{i\in I}$ is called the landmark set and $X$ is the witness set.
When the witness set coincides with the landmarks, we thus define a non-empty
intersection inside the sample set $(x_{i})_{i\in I}$ as the criterion for a
simplex. We also define the _Vietoris-Rips complex_ $VR_{r}((x_{i}),X)$
containing a $q$-simplex whenever

$B(x_{i},r)\cap B(x_{j},r)\neq\emptyset\ \text{ for all }i,j\in I.$

The two structures are not as different as they might appear, as the
difference between the criteria for spanning a simplex is whether the vertex
set is contained in a ball of radius or of diameter $r$.
The principle of the important topological data analysis scheme of _persistent
homology_ then is to record how the homology of these complexes varies as a
function of $r$. [19, 42, 18, 12]
Of course, every simplex of the Čech complex is also a simplex of the
Vietoris-Rips complex, but not necessarily conversely unless for each simplex
at least one of the balls of diameter $r$ containing the vertex set of that
simplex has a center in the witness set.
Deviation from hyperconvexity lets the Vietoris-Rips complex contain more
simplices than the Čech complex, or conversely

######  Lemma 5.1.

In a hyperconvex space, all simplices that are filled in the Vietoris-Rips
complex are also filled in the Čech complex. In particular, there is no
contribution to local homology from unfilled simplices. ∎

For instance, we can take a sample $(x_{i})_{i\in I}$ from a geodesic metric
space $(X,d)$ and compare $VR_{r}((x_{i}),X)$ with
$\check{C}_{r}((x_{i}),E(X))$. For the latter complex, we take the hyperconvex
hull of $X$, i.e. $E(X)$, as the witness set. It is clear that
$\check{C}_{r}((x_{i}),E(X))\subset VR_{r}((x_{i}),X)$, as $X$ is a geodesic
space and hence totally convex. Conversely, every simplex in
$VR_{r}((x_{i}),X)$ is defined according to the criterion that balls of radius
$r$ around its vertices intersect pairwise, which by hyperconvexity of $E(X)$
implies the existence of a common point between them in $E(X)$. In other
words, the Vietoris-Rips complex of a metric family $(x_{i})_{i\in I},d$
coincides with its Čech complex but with different witness sets. This natural
principle has been used in [36] to study the metric thickening of $S^{1}$ in
its hyperconvex hull. A thorough study of the Čech and the Vietoris-Rips
filtration of $S^{1}$ can be found in [2, 1].
If $X$ is a closed Riemannian manifold, for small-enough radius $r$ depending
on the injectivity radius and a curvature bound, $VR_{r}(X)$ is homotopy
equivalent to $X$ by a well known theorem of Hausmann [27]. On the other hand
according to the nerve lemma, whenever $X$ is a paracompact space and the
family of open balls around sample points $(x_{i})_{i\in I}$ with radius $r>0$
define a cover such that the non-empty intersections of any finite number of
them is contractible, the Čech complex $\check{C}_{<r}((x_{i}),X)$ is homotopy
equivalent to the original space $X$, c.f. [26]. Although Hausmann’s theorem
is restricted to the case where the original space, from which the sample is
taken, is a Riemannian manifold, both construction at some point reveal the
topology of the space. However, the Vietoris-Rips filtration ignores the
geometry of the space beyond the pairwise relations. The extent to which
higher order relations are overlooked by considering Vietoris-Rips complexes
can be quantified by computing the deviation from hyperconvexity of different
orders. This measures how much one must expand balls to obtain a simplex in
the Čech complex of $(x_{i})$ with witness set $X$ after that simplex is
observed in the Čech complex of $(x_{i})$ with witness set $E(X)$. The upper
bound $2$ for this scale is usually stated in the TDA literature, but this
bound is not sharp.
For instance, let us consider equilateral triangles of perimeter $3a$ in the
Euclidean plane, in a circle and in a metric tree. That is,
$(x_{1},x_{2},x_{3})$, $(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3})$ and
$(\bar{x}_{1},\bar{x}_{2},\bar{x}_{3})$ are comparison triangles in the
Euclidean plane, a circle and a hyperconvex space, respectively. As noted in
(3), $r=\frac{a}{2}$ is the radius at which each of these triples forms a
simplex in the corresponding Vietoris-Rips complex. However, we only need the
upper bound of $2$ in the case of
$(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3})$, where the point are sampled
from a circle which has the highest deviation from hyperconvexity, for
expanding the balls to obtain the simplex in the Čech complex, c.f. [31].
One can also more generally let the radii of the balls be different. That is,
for a vertex set $(x_{i})_{i\in I}$ and a corresponding non-negative radius
function $r$, we define the Čech complex containing a $q$-simplex
$x_{1},...,x_{q+1}$ whenever

$\bigcap_{i=1,\dots,q+1}B(x_{i},r(x_{i}))\neq\emptyset.$

The Vietoris-Rips complex is defined in a similar way. And one can then look
at the resulting constructions for all such radius functions simultaneously
[31].

##  6 Curvature

We can use the preceding concepts to compare spaces with each other, or with
reference spaces, like Euclidean space. In geometry, such a comparison is
quantified by the concept of curvature. From our abstract perspective,
curvature relates intersection patterns of balls to convexity properties of
distance functions.

As pointed out by Klingenberg [34], the beginning of the theory of spaces of
negative curvature can be dated to the work of von Mangoldt [41] in 1881 who
showed that on a complete simply connected surface of negative curvature,
geodesics starting at the same point diverge and can never meet again. This
implies that the exponential map is a diffeomorphism. Apparently unaware of
von Mangoldt’s work, Hadamard [25] in 1898 proved further results about
geodesics on surfaces of negative curvature. E.Cartan [13] later considered
negatively curved Riemannian manifolds of any dimension. For our purposes,
non-positive, as opposed to negative, curvature is the appropriate concept, as
we are interested in comparison theorems.

Let us first recall a by now classical concept of non-positive curvature,
introduced by Alexandrov [4].

######  Definition 6.1.

The geodesic space $(X,d)$ is a $CAT(0)$-space if for all geodesics
$c_{1},c_{2}:[0,1]\longrightarrow X$ with $c_{1}(0)=c_{2}(0)$

$d(c_{1}(t),c_{2}(s))\leq\|{\bar{c}_{1}(t)-\bar{c}_{2}(s)}\|,\,\forall\
t,s\in[0,1]$ (11)

where $\bar{c}_{1},\bar{c}_{2}:[0,1]\longrightarrow\mathbb{R}^{2}$ are the
sides of the Euclidean comparison triangle in $\mathbb{R}^{2}$ with the same
side lengths as the triangle $\bigtriangleup(c_{1}(0),c_{1}(1),c_{2}(1))$.

According to this definition, triangles in $CAT(0)$-spaces are not thicker
than Euclidean triangles with the same side lengths, c.f. [32, 10, 9, 3]

There is another important concept of non-positive curvature, introduced by
Busemann [11].

######  Definition 6.2.

A geodesic space $(X,d)$ is a Busemann convex space if for every two geodesics
$c_{1},c_{2}:[0,1]\longrightarrow X$ with $c_{1}(0)=c_{2}(0)$, the distance
function $t\mapsto d(c_{1}(t),c_{2}(t))$ is convex.

Geodesics in Busemann space diverge at least as fast as in Euclidean space.
Every $CAT(0)$ space is Busemann convex but not conversely. For complete
Riemannian manifolds, however, the two definitions agree and are equivalent to
non-positive sectional curvature in the sense of Riemann.
Several generalizations of these definitions to metric spaces that are not
necessarily geodesic have been proposed, for instance [7, 8, 3]. We now
present our definition from [30].

######  Definition 6.3.

The metric space $(X,d)$ has _non-positive curvature_ if for each triple
$(x_{1},x_{2},x_{3})$ in $X$ with the comparison triangle
$\bigtriangleup(\bar{x}_{1},\bar{x}_{2},\bar{x}_{3})$ in ${\mathbb{R}}^{2}$,
one has

$\rho(x_{1},x_{2},x_{3})\leq\rho(\bar{x}_{1},\bar{x}_{2},\bar{x}_{3}),$

where $\rho(\bar{x}_{1},\bar{x}_{2},\bar{x}_{3})$ is similarly defined by

$\rho(\bar{x}_{1},\bar{x}_{2},\bar{x}_{3}):=\min_{x\in{\mathbb{R}}^{2}}\max_{i=1,2,3}\dfrac{\|x-\bar{x}_{i}\|}{r_{i}}.$

According to this definition, the circumcenter of a triangle in a non-
positively curved space is at least as close to the vertices as in the
Euclidean case. In other words, there is chance of finding a better
intermediate point for each triple of points in such a space than in Euclidean
plane.
For any triple of closed balls $\\{B(x_{i},r_{i});\>i=1,2,3\\}$ with pairwise
intersection, $\bigcap_{i=1,2,3}B(x_{i},\rho r_{i})$ is non-empty whenever
$B(\bar{x}_{i},\rho r_{i})$, $i=1,2,3$, have a common point. Thus, balls do
not need to be enlarged more than in Euclidean case to get triple
intersection. Thus, we can again formulate a

######  Principle 6.1.

Balls intersect at least as easily as in Euclidean space.

Examples:

  * •

Tripod spaces have non-positive curvature in the sense of Def. 6.3, because
there, $\rho=1$, which is the smallest possible value.

  * •

Complete $CAT(0)$ spaces have non-positive curvature in the sense of Def. 6.3.
The converse not true; in fact, our spaces need not be geodesic, nor have
unique geodesics.

  * •

Approximate version applies to discrete spaces. This is obviously important
for questions of data analysis, and this in fact constitutes one of the
motivations for Def. 6.3.

We also have

######  Theorem 6.1.

A complete Riemannian manifold $(N,g)$ has non-positive curvature iff it has
non-positive sectional curvature, c.f. [30].

Obviously, with the same concepts and constructions, one can also define upper
curvature bounds other than 0, by comparison with suitably scaled 2-spheres or
hyperbolic planes.

##  7 Conclusions

The Čech construction assigns to a cover $\mathcal{U}=(U_{i})_{i\in I}$ of $X$
a simplicial complex $\Sigma(\mathcal{U})$ with vertex set $I$ and a simplex
$\sigma_{J}$ whenever $\bigcap_{j\in J}U_{j}\neq\emptyset$ for $J\subset
I$.When all intersections are contractible, the homology of
$\Sigma(\mathcal{U})$ equals that of $X$ (under some rather general
topological conditions on $X$). When $(X,d)$ is metric space, we can use
covers by (open or closed) distance balls. Now, when $(X,d)$ is a hyperconvex
metric space, and if we use a cover $\mathcal{U}$ by distance balls, then
whenever

$\bigcap_{j\in J\backslash\\{j_{0}\\}}U_{j}\neq\emptyset\text{ for every
}j_{0}\in J,$ (12)

then also

$\bigcap_{j\in J}U_{j}\neq\emptyset,$ (13)

i.e., whenever $\Sigma(\mathcal{U})$ contains all the boundary facets of some
simplex, it also contains that simplex itself. It even satisfies the stronger
condition that whenever $\Sigma(\mathcal{U})$ contains all the boundary faces
of dimension $1$ of some simplex, it also contains that simplex itself. This
means that $\Sigma(\mathcal{U})$ is a flag complex. Thus, there are no holes
of the type of unfilled simplices, and no corresponding contributions to
homology groups.

As hyperconvex spaces are contractible, then whenever non-trivial homology
groups arise in Čech filtrations, the space cannot be hyperconvex, but only
$\lambda$-hyperconvex for some $\lambda>1$. But every complete metric space is
$\lambda$-hyperconvex for some $1\leq\lambda\leq 2$, c.f. [24]. (In the
discrete case, one might work also with $\delta$-hyperbolicity for
$\delta>0$.)
From that perspective, hyperconvex spaces are the simplest model spaces, and
homology can be seen as a topological measure for the deviation from such a
model. However, this geometric interpretation has been dismissed in
topological data analysis, by considering the Vietoris-Rips filtration instead
of Čech, for the benefit of reducing computational complexity. Still, it is
possible to infer topological information about a space from the Vietoris-Rips
filtration, based on Hausmann’s theorem. However, when one samples a metric
space, this depends on how dense sample is and the results are accurate only
for small radii. For instance, the Vietoris-Rips complexes of $S^{1}$ admit
holes of dimension larger than $1$ as the radius increases, c.f. [1].

Homology groups, and Betti numbers as integer invariants are fundamental
topological invariants. Geometry can provide more refined real valued
invariants. And after Riemann [38, 39], the fundamental geometric invariants
are curvatures. In our framework, the essential geometric content of curvature
can be extracted for general metric spaces. The basic class of model spaces
for curvature is given by the tripod spaces, a special class containing
hyperconvex spaces. From that perspective, the geometric content of curvature
in the abstract setting considered here is the deviation from the tripod
condition. Euclidean spaces only have a subsidiary role, based on a
normalization of curvature that assigns the value $0$ to them.
Considering Euclidean spaces as model spaces is traditionally justified by the
fact that spaces whose universal cover has synthetic curvature $\leq 0$ in the
sense of Alexandrov are homotopically trivial in the sense that their higher
homotopy groups vanish. In technical terms, they are $K(\pi,1)$ spaces, with
$\pi$ standing for the first homotopy group. The perspective developed here,
however, is a homological and not a homotopical one, and therefore, our
natural comparison spaces are tripods. We have started their investigation in
[30, 31]. A more systematic investigation of their properties should be of
interest.
In order to get stronger topological properties, like those of hyperconvex
spaces, which are homologically trivial, we might need conditions involving
collections of more than three points.
In fact, according [37, Theorem 4.2], If $X$ is a tripod Banach space on which
every collection of four closed balls $\\{B(x_{i},r_{i})\\}_{i=1}^{4}$ with
non-empty pairwise intersection has a non-void intersection, then every finite
family of closed balls with non-empty pairwise intersection has also a non-
trivial intersection. In this case, the Vietoris-Rips and Čech complexes
coincide.

One can also think about higher order relations and how they can be obtained
from sub-relations (that is from the relations existing in all subsets of some
smaller size). For instance, in some metric spaces, a family of $n$ balls has
a common point if every subfamily of size $k$ in it has a non-empty
intersection. [37] calls this property the $(n,k)$-intersection property. For
instance, Helly’s theorem says that Euclidean space ${\mathbb{R}}^{d}$ has the
$(n,d+1)$-intersection property for $n\geq d+2$. For a given metric space, one
can compute the deviation from such a property.
From the perspective of Čech complexes, this deviation could be quantified by
the scaling parameter needed to fill an $(n-1)$-simplex after all the faces of
dimension $k-1$ are filled. The quantitative measure we introduced provides us
with the scaling function to fill a $2$-simplex after its $1$-dimensional
boundary faces are filled.

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