The Cardinality of a Metric Space

The Cardinalityof a

Metric Space

Tom Leinster (Glasgow/EPSRC)

Parts joint with Simon Willerton (Sheffield)

Where does the idea come from?

finite

categories

finite

enriched categories

finite

metric spaces

⊂ ⊂

every finite category Ahas a cardinality

(or Euler characteristic)|A|

But where does this idea come from?See two papers

listed on my web page

�

every finite enriched category Ahas a cardinality |A|

*

GEN

ERALI

ZE

every finite metric space Ahas a cardinality |A|

?

SPEC

IALIZE


finite categories

finite

enriched categories

finite

metric spaces

⊂ ⊂





�


*

GEN

ERALI

ZE


?

SPEC

IALIZE


finite categories

finite

enriched categories

finite

metric spaces

⊂ ⊂





�


*

GEN

ERALI

ZE


?

SPEC

IALIZE


finite categories

finite

enriched categories

finite

metric spaces

⊂ ⊂





�


*

GEN

ERALI

ZE


?

SPEC

IALIZE


finite categories

finite enriched categories

finite

metric spaces

⊂ ⊂





�


*

GEN

ERALI

ZE


?

SPEC

IALIZE


finite categories


finite metric spaces

⊂ ⊂





�


*

GEN

ERALI

ZE


?

SPEC

IALIZE


finite categories


finite metric spaces

⊂ ⊂





�


*

GEN

ERALI

ZE


?

SPEC

IALIZE

1. The cardinality of a finite metric space

DefinitionLet A = {a1, . . . , an} be a finite metric space.Write Z for the n × n matrix with Zij = e−2d(ai ,aj ).

The cardinality of A is

|A| =∑i ,j

(Z−1)ij ∈ R.

RemarkIn principle, Z is defined by Zij = Cd(ai ,aj ) for some constant C .We’ll see that taking C = e−2 is most convenient.

Warning (Tao)

There exist finite metric spaces whose cardinality is undefined(i.e. with Z non-invertible).


DefinitionLet A = {a1, . . . , an} be a finite metric space.Write Z for the n × n matrix with Zij = e−2d(ai ,aj ).The cardinality of A is

|A| =∑i ,j

(Z−1)ij ∈ R.


Warning (Tao)




|A| =∑i ,j

(Z−1)ij ∈ R.


Warning (Tao)




|A| =∑i ,j

(Z−1)ij ∈ R.


Warning (Tao)


Reference

‘Metric spaces’, post at The n-Category Cafe, 9 February 2008



|A| =∑i ,j

(Z−1)ij ∈ R.

RemarkIn principle Z is defined by Zij = Cd(ai ,aj ) for some constant C .We’ll see later that taking C = e−2 is most convenient.

Warning (Tao)

There exist finite metric spaces whose cardinality is undefined(i.e. Z is not invertible).


Example (two-point spaces)

Let A = (•← d →•). Then

Z =

(e−2·0 e−2·d

e−2·d e−2·0

)=

(1 e−2d

e−2d 1

),

Z−1 =1

1− e−4d

(1 −e−2d

−e−2d 1

),

|A| =1

1− e−4d(1− e−2d − e−2d + 1) = 1 + tanh(d) .

0

1

2

|A|

d



Let A = (•← d →•). Then

Z =

(e−2·0 e−2·d

e−2·d e−2·0

)=

(1 e−2d

e−2d 1

),

Z−1 =1

1− e−4d

(1 −e−2d

−e−2d 1

),

|A| =1

1− e−4d(1− e−2d − e−2d + 1) = 1 + tanh(d) .

0

1

2

|A|

d



Let A = (•← d →•). Then

Z =

(e−2·0 e−2·d

e−2·d e−2·0

)=

(1 e−2d

e−2d 1

),

Z−1 =1

1− e−4d

(1 −e−2d

−e−2d 1

),

|A| =1

1− e−4d(1− e−2d − e−2d + 1) = 1 + tanh(d) .

0

1

2

|A|

d



Let A = (•← d →•). Then

Z =

(e−2·0 e−2·d

e−2·d e−2·0

)=

(1 e−2d

e−2d 1

),

Z−1 =1

1− e−4d

(1 −e−2d

−e−2d 1

),

|A| =1

1− e−4d(1− e−2d − e−2d + 1) = 1 + tanh(d) .

0

1

2

|A|

d


Cardinality assigns to each metric space not just a number, but a function.

DefinitionGiven t ∈ (0,∞), write tA for A scaled up by a factor of t.The cardinality function of A is the partial function

χA : (0,∞) −→ R, t 7−→ |tA|.




χA : (0,∞) −→ R, t 7−→ |tA|.




χA : (0,∞) −→ R, t 7−→ |tA|.

Example

A = (•← 1→•):

0

1

2

χA(t)

t




χA : (0,∞) −→ R, t 7−→ |tA|.

Generic example

0

no. points of A

χA(t)

t

︸︷︷︸wild

︸︷︷︸increasing

2. Some geometric measure theoryRef: Schanuel, ‘What is the length of a potato?’

Suppose we want a ruler of length 1 cm: 0 1 .

A half-open interval is good:

• ◦1 cm

∪ • ◦1 cm = • ◦2 cm

A closed interval is not so good:

• •1 cm

∪ • •1 cm = • •2 cm+ •

So we declare:

size([0, 1]) = 1 cm + 1 point = 1 cm1 + 1 cm0 = 1 cm + 1.

In general,size([0, `]) = ` cm + 1.




• ◦1 cm

∪ • ◦1 cm = • ◦2 cm


• •1 cm

∪ • •1 cm = • •2 cm+ •

So we declare:






• ◦1 cm

∪ • ◦1 cm = • ◦2 cm


• •1 cm

∪ • •1 cm = • •2 cm+ •

So we declare:






• ◦1 cm

∪ • ◦1 cm = • ◦2 cm


• •1 cm

∪ • •1 cm = • •2 cm+ •

So we declare:






• ◦1 cm

∪ • ◦1 cm = • ◦2 cm


• •1 cm

∪ • •1 cm = • •2 cm+ •

So we declare:



2. Some geometric measure theory

Examples

• Size of rectangle`

k

is

(k cm + 1)(` cm + 1) = k` cm2 + (k + `) cm + 1.

• Size of hollow trianglek `

m

is

(k cm + 1) + (` cm + 1) + (m cm + 1)− 3 = (k + ` + m) cm + 0.

• Similarly, can compute sizes of , , , . . .


Examples


k

is

(k cm + 1)(` cm + 1) = kH

area

` cm2 + (kH

12×perimeter

+ `) cm +H

Euler char

1.


m

is

(k cm + 1) + (` cm + 1) + (m cm + 1)− 3 = (k + ` + m) cm + 0.



Examples


k

is

(k cm + 1)(` cm + 1) = kH

area

` cm2 + (kH

12×perimeter

+ `) cm +H

Euler char

1.


m

is

(k cm + 1) + (` cm + 1) + (m cm + 1)− 3 = (k + ` + m) cm

+ 0.



Examples


k

is

(k cm + 1)(` cm + 1) = kH

area

` cm2 + (kH

12×perimeter

+ `) cm +H

Euler char

1.


m

is

(k cm + 1) + (` cm + 1) + (m cm + 1)− 3 = (k +H

perimeter

` + m) cm +H

Euler char

0.



Examples


k

is

(k cm + 1)(` cm + 1) = kH

area

` cm2 + (kH

12×perimeter

+ `) cm +H

Euler char

1.


m

is

(k cm + 1) + (` cm + 1) + (m cm + 1)− 3 = (k +H

perimeter

` + m) cm +H

Euler char

0.



Fix n ∈ N. What ‘measures’ can be defined on the ‘nice’ subsets of Rn?

Hadwiger’s Theorem says that there are essentially n + 1 such measures.They are called the intrinsic volumes,

µ0, µ1, . . . , µn,

and µd is d-dimensional: µd(tA) = tdµd(A).

Example

When n = 2, have three measures:

µ0 =

Euler characteristic

µ1 =

perimeter

µ2 =

area.




µ0, µ1, . . . , µn,


Example


µ0 =


µ1 =

perimeter

µ2 =

area.




µ0, µ1, . . . , µn,


Example


µ0 =


µ1 =

perimeter

µ2 =

area.




µ0, µ1, . . . , µn,


Example


µ0 = Euler characteristic

µ1 = perimeter

µ2 = area.

3. The cardinality of a compact metric space

IdeaGiven a compact metric space A, choose a sequence

A0 ⊆ A1 ⊆ · · ·

of finite subsets of A, with⋃

i Ai dense in A.

Try to define

|A| = limi→∞|Ai |.

TheoremLet A = [0, `] and take any sequence (Ai ) as above. Then

limi→∞|Ai | = ` + 1.

Remark



A0 ⊆ A1 ⊆ · · ·


i Ai dense in A. Try to define

|A| = limi→∞|Ai |.


limi→∞|Ai | = ` + 1.

Remark



A0 ⊆ A1 ⊆ · · ·



|A| = limi→∞|Ai |.


limi→∞|Ai | =

` + 1.

Remark



A0 ⊆ A1 ⊆ · · ·



|A| = limi→∞|Ai |.


limi→∞|Ai | = ` + 1.

Remark



A0 ⊆ A1 ⊆ · · ·



|A| = limi→∞|Ai |.


limi→∞|Ai | = ` + 1.

RemarkThis is the reason for the choice of the constant e−2.



A0 ⊆ A1 ⊆ · · ·



|A| = limi→∞|Ai |.


limi→∞|Ai | = ` + 1.

Remark[0, `] has cardinality function t 7→ | [0, t`] | = `t + 1: so ‘t = cm’.


Assume now that all spaces mentioned have well-defined cardinality.

ProductsLet A and B be compact metric spaces. Then

|A× B| = |A| · |B|

as long as we give A× B the ‘d1 metric’:

d((a, b), (a′, b′)) = d(a, a′) + d(b, b′).

Example (rectangle)

With this metric, [0, k]× [0, `] =`

k

has cardinality function

t 7→ (kt + 1)(`t + 1) = k` t2 + (k + `)t + 1.




|A× B| = |A| · |B|


d((a, b), (a′, b′)) = d(a, a′) + d(b, b′).

Example (rectangle)


k


t 7→ (kt + 1)(`t + 1) = k` t2 + (k + `)t + 1.




|A× B| = |A| · |B|


d((a, b), (a′, b′)) = d(a, a′) + d(b, b′).

Example (rectangle)


k


t 7→ (kt + 1)(`t + 1) = k` t2 + (k + `)t + 1.




|A× B| = |A| · |B|


d((a, b), (a′, b′)) = d(a, a′) + d(b, b′).

Example (rectangle)


k


t 7→ (kt + 1)(`t + 1) = k` t2 + (k + `)t + 1.


Example (circle)

Let C` be the circle of circumference `, with metricgiven by length of shortest arc.

a

b

d(a, b)

Then

|C`| =`

1− e−`=

∞∑n=0

Bn(−`)n

n!

where Bn is the nth Bernoulli number.

Asymptotics

• |C`| → 1 as `→ 0.

• |C`| − `→ 0 as `→∞: so when ` is large, |C`| ≈H

perimeter

` +H

Euler char

0.


Example (circle)


a

b

d(a, b)

Then

|C`| =`

1− e−`=

∞∑n=0

Bn(−`)n

n!


Asymptotics

• |C`| → 1 as `→ 0.


perimeter

` +H

Euler char

0.


Example (circle)


a

b

d(a, b)

Then

|C`| =`

1− e−`=

∞∑n=0

Bn(−`)n

n!


Asymptotics

• |C`| → 1 as `→ 0.


perimeter

` +H

Euler char

0.


Example (circle)


a

b

d(a, b)

Then

|C`| =`

1− e−`=

∞∑n=0

Bn(−`)n

n!


Asymptotics

• |C`| → 1 as `→ 0.

• |C`| − `→ 0 as `→∞

: so when ` is large, |C`| ≈H

perimeter

` +H

Euler char

0.


Example (circle)


a

b

d(a, b)

Then

|C`| =`

1− e−`=

∞∑n=0

Bn(−`)n

n!


Asymptotics

• |C`| → 1 as `→ 0.


perimeter

` +H

Euler char

0.


Hypothesis

All the important invariants of compact metric spacescan be derived from cardinality

Examples

• Euler characteristic

• Intrinsic volumes µ0, µ1, . . .

• Hausdorff dimension


Hypothesis


Examples





Hypothesis


Examples





Hypothesis


Examples




Review

metric spacesas enriched categories

cardinality (Euler char)of finite categories

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cardinality offinite metric spaces

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cardinality ofcompact metric spaces

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invariants fromgeometric measure theory

Review



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Review



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How do you quantify the diversity of an ecosystem?

Hundreds of numerical measures of biodiversity have been proposed.

diversity:

Example of a diversity measure

‘Effective number of species’

‘Measuring biological diversity’,Andrew SolowStephen PolaskyEnvironmental and Ecological Statistics 1 (1994), 95–107.



diversity:






diversity: low






diversity: higher






diversity:






diversity:



‘Measuring biological diversity’,Andrew Solow,Stephen Polasky,Environmental and Ecological Statistics 1 (1994), 95–107.



diversity:



‘Measuring biological diversity’,Andrew Solow (Marine Policy Center, Woods Hole),Stephen Polasky (Agricultural and Resource Economics, Oregon State),Environmental and Ecological Statistics 1 (1994), 95–107.



diversity:


‘Effective number of species’ = cardinality of the metric space of species

‘Measuring biological diversity’,Andrew Solow (Marine Policy Center, Woods Hole),Stephen Polasky (Agricultural and Resource Economics, Oregon State),Environmental and Ecological Statistics 1 (1994), 95–107.

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The Cardinality of a Metric Space