Integral geometry for the 1-norm

Tom Leinster

Glasgow/EPSRC

arXiv:1012.5881v2

Plan

1. Introduction

2. Convex geometry in the 1-norm

3. Integral geometry in the 1-norm

Plan

1. Introduction

2. Convex geometry in the 1-norm

3. Integral geometry in the 1-norm

Plan

1. Introduction

2. Convex geometry in the 1-norm

3. Integral geometry in the 1-norm

Plan

1. Introduction

2. Convex geometry in the 1-norm

3. Integral geometry in the 1-norm

1. Introduction

Beyond Euclidean space

Integral geometry was originally developed in Euclidean space.

It has been extended in many ways:

• to manifolds: Alesker, Fu, . . .

• to Minkowski spaces and projective Finsler spaces:Alvarez Paiva, Bernig, Fernandes, Schneider, Wieacker, . . .

• to complex spaces: Alesker, Bernig, Fu, Solanes, . . .

• . . .

We’ll consider integral geometry in metric spaces.

Beyond Euclidean space

Integral geometry was originally developed in Euclidean space.

It has been extended in many ways:

• to manifolds: Alesker, Fu, . . .

• to Minkowski spaces and projective Finsler spaces:Alvarez Paiva, Bernig, Fernandes, Schneider, Wieacker, . . .

• to complex spaces: Alesker, Bernig, Fu, Solanes, . . .

• . . .

We’ll consider integral geometry in metric spaces.

Beyond Euclidean space

Integral geometry was originally developed in Euclidean space.

It has been extended in many ways:

• to manifolds: Alesker, Fu, . . .

• to Minkowski spaces and projective Finsler spaces:Alvarez Paiva, Bernig, Fernandes, Schneider, Wieacker, . . .

• to complex spaces: Alesker, Bernig, Fu, Solanes, . . .

• . . .

We’ll consider integral geometry in metric spaces.

Beyond Euclidean space

Integral geometry was originally developed in Euclidean space.

It has been extended in many ways:

• to manifolds: Alesker, Fu, . . .

• to Minkowski spaces and projective Finsler spaces:Alvarez Paiva, Bernig, Fernandes, Schneider, Wieacker, . . .

• to complex spaces: Alesker, Bernig, Fu, Solanes, . . .

• . . .

We’ll consider integral geometry in metric spaces.

Beyond Euclidean space

Integral geometry was originally developed in Euclidean space.

It has been extended in many ways:

• to manifolds: Alesker, Fu, . . .

• to Minkowski spaces and projective Finsler spaces:Alvarez Paiva, Bernig, Fernandes, Schneider, Wieacker, . . .

• to complex spaces: Alesker, Bernig, Fu, Solanes, . . .

• . . .

We’ll consider integral geometry in metric spaces.

Beyond Euclidean space

Integral geometry was originally developed in Euclidean space.

It has been extended in many ways:

• to manifolds: Alesker, Fu, . . .

• to Minkowski spaces and projective Finsler spaces:Alvarez Paiva, Bernig, Fernandes, Schneider, Wieacker, . . .

• to complex spaces: Alesker, Bernig, Fu, Solanes, . . .

• . . .

We’ll consider integral geometry in metric spaces.

Beyond Euclidean space

Integral geometry was originally developed in Euclidean space.

It has been extended in many ways:

• to manifolds: Alesker, Fu, . . .

• to Minkowski spaces and projective Finsler spaces:Alvarez Paiva, Bernig, Fernandes, Schneider, Wieacker, . . .

• to complex spaces: Alesker, Bernig, Fu, Solanes, . . .

• . . .

We’ll consider integral geometry in metric spaces.

Integral geometry in metric spaces

Typical goal: imitate Hadwiger’s theorem in some metric space A 6= Rn.

• X ⊆ A is geodesic if for all x , y ∈ X , there exists adistance-preserving path γ : [0, d(x , y)] → X joining x and y .

• A valuation on A is a function

{compact geodesic subsets of A} → R

satisfying the usual equations (where defined).

• Valuations on A form a vector space,

Val(A) = {continuous isometry-invariant valuations on A}.

E.g. Val(Rn) ∼= Rn+1.

Every metric space A presents a challenge:

What is Val(A)?

Integral geometry in metric spaces

Typical goal: imitate Hadwiger’s theorem in some metric space A 6= Rn.

• X ⊆ A is geodesic if for all x , y ∈ X , there exists adistance-preserving path γ : [0, d(x , y)] → X joining x and y .

• A valuation on A is a function

{compact geodesic subsets of A} → R

satisfying the usual equations (where defined).

• Valuations on A form a vector space,

Val(A) = {continuous isometry-invariant valuations on A}.

E.g. Val(Rn) ∼= Rn+1.

Every metric space A presents a challenge:

What is Val(A)?

Integral geometry in metric spaces

Typical goal: imitate Hadwiger’s theorem in some metric space A 6= Rn.

• X ⊆ A is geodesic if for all x , y ∈ X , there exists adistance-preserving path γ : [0, d(x , y)] → X joining x and y .

• A valuation on A is a function

{compact geodesic subsets of A} → R

satisfying the usual equations (where defined).

• Valuations on A form a vector space,

Val(A) = {continuous isometry-invariant valuations on A}.

E.g. Val(Rn) ∼= Rn+1.

Every metric space A presents a challenge:

What is Val(A)?

Integral geometry in metric spaces

Typical goal: imitate Hadwiger’s theorem in some metric space A 6= Rn.

• X ⊆ A is geodesic if for all x , y ∈ X , there exists adistance-preserving path γ : [0, d(x , y)] → X joining x and y .

• A valuation on A is a function

{compact geodesic subsets of A} → R

satisfying the usual equations (where defined).

• Valuations on A form a vector space,

Val(A) = {continuous isometry-invariant valuations on A}.

E.g. Val(Rn) ∼= Rn+1.

Every metric space A presents a challenge:

What is Val(A)?

Integral geometry in metric spaces

Typical goal: imitate Hadwiger’s theorem in some metric space A 6= Rn.

• X ⊆ A is geodesic if for all x , y ∈ X , there exists adistance-preserving path γ : [0, d(x , y)] → X joining x and y .

• A valuation on A is a function

{compact geodesic subsets of A} → R

satisfying the usual equations (where defined).

• Valuations on A form a vector space,

Val(A) = {continuous isometry-invariant valuations on A}.

E.g. Val(Rn) ∼= Rn+1.

Every metric space A presents a challenge:

What is Val(A)?

Integral geometry in metric spaces

Typical goal: imitate Hadwiger’s theorem in some metric space A 6= Rn.

• X ⊆ A is geodesic if for all x , y ∈ X , there exists adistance-preserving path γ : [0, d(x , y)] → X joining x and y .

• A valuation on A is a function

{compact geodesic subsets of A} → R

satisfying the usual equations (where defined).

• Valuations on A form a vector space,

Val(A) = {continuous isometry-invariant valuations on A}.

E.g. Val(Rn) ∼= Rn+1.

Every metric space A presents a challenge:

What is Val(A)?

Integral geometry in metric spaces

Typical goal: imitate Hadwiger’s theorem in some metric space A 6= Rn.

• X ⊆ A is geodesic if for all x , y ∈ X , there exists adistance-preserving path γ : [0, d(x , y)] → X joining x and y .

• A valuation on A is a function

{compact geodesic subsets of A} → R

satisfying the usual equations (where defined).

• Valuations on A form a vector space,

Val(A) = {continuous isometry-invariant valuations on A}.

E.g. Val(Rn) ∼= Rn+1.

Every metric space A presents a challenge:

What is Val(A)?

Rn with the p-norm

Write `np for Rn with metric d(x , y) =

(∑|xi − yi |p

)1/p(1 ≤ p < ∞).

1 2 p

groupisometrygeodesic

sets

1 2 p

groupisometrygeodesic

sets

dim(Val(`n2)) = n + 1

H

dim(Val(`n1)) = ?

H

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

dim(Val(`np)) = ∞

p 6= 1, 2:

• p 6= 1: geodesic ⇐⇒ convex

• p 6= 2: only isometries are translations, coordinate permutations, andreflections in coordinate hyperplanes

Rn with the p-norm

Write `np for Rn with metric d(x , y) =

(∑|xi − yi |p

)1/p(1 ≤ p < ∞).

1 2 p

groupisometrygeodesic

sets

1 2 p

groupisometrygeodesic

sets

dim(Val(`n2)) = n + 1

H

dim(Val(`n1)) = ?

H

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

dim(Val(`np)) = ∞

p 6= 1, 2:

• p 6= 1: geodesic ⇐⇒ convex

• p 6= 2: only isometries are translations, coordinate permutations, andreflections in coordinate hyperplanes

Rn with the p-norm

Write `np for Rn with metric d(x , y) =

(∑|xi − yi |p

)1/p(1 ≤ p < ∞).

1 2 p

groupisometrygeodesic

sets

1 2 p

groupisometrygeodesic

sets

dim(Val(`n2)) = n + 1

H

dim(Val(`n1)) = ?

H

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

dim(Val(`np)) = ∞

p 6= 1, 2:

• p 6= 1: geodesic ⇐⇒ convex

• p 6= 2: only isometries are translations, coordinate permutations, andreflections in coordinate hyperplanes

Rn with the p-norm

Write `np for Rn with metric d(x , y) =

(∑|xi − yi |p

)1/p(1 ≤ p < ∞).

1 2 p

groupisometrygeodesic

sets

1 2 p

groupisometrygeodesic

sets

dim(Val(`n2)) = n + 1

H

dim(Val(`n1)) = ?

H

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

dim(Val(`np)) = ∞

p 6= 1, 2:

• p 6= 1: geodesic ⇐⇒ convex

• p 6= 2: only isometries are translations, coordinate permutations, andreflections in coordinate hyperplanes

Rn with the p-norm

Write `np for Rn with metric d(x , y) =

(∑|xi − yi |p

)1/p(1 ≤ p < ∞).

1 2 p

groupisometrygeodesic

sets

1 2 p

groupisometrygeodesic

sets

dim(Val(`n2)) = n + 1

H

dim(Val(`n1)) = ?

H

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

dim(Val(`np)) = ∞

p 6= 1, 2:

• p 6= 1: geodesic ⇐⇒ convex

• p 6= 2: only isometries are translations, coordinate permutations, andreflections in coordinate hyperplanes

Rn with the p-norm

Write `np for Rn with metric d(x , y) =

(∑|xi − yi |p

)1/p(1 ≤ p < ∞).

1 2 p

groupisometrygeodesic

sets

1 2 p

groupisometrygeodesic

sets

dim(Val(`n2)) = n + 1

H

dim(Val(`n1)) = ?

H

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

dim(Val(`np)) = ∞

p 6= 1, 2:

• p 6= 1: geodesic ⇐⇒ convex

• p 6= 2: only isometries are translations, coordinate permutations, andreflections in coordinate hyperplanes

Rn with the p-norm

Write `np for Rn with metric d(x , y) =

(∑|xi − yi |p

)1/p(1 ≤ p < ∞).

1 2 p

groupisometrygeodesic

sets

1 2 p

groupisometrygeodesic

sets

dim(Val(`n2)) = n + 1

H

dim(Val(`n1)) = ?

H

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

dim(Val(`np)) = ∞

p 6= 1, 2:

• p 6= 1: geodesic ⇐⇒ convex

• p 6= 2: only isometries are translations, coordinate permutations, andreflections in coordinate hyperplanes

Rn with the p-norm

Write `np for Rn with metric d(x , y) =

(∑|xi − yi |p

)1/p(1 ≤ p < ∞).

1 2 p

groupisometrygeodesic

sets

1 2 p

groupisometrygeodesic

sets

dim(Val(`n2)) = n + 1

H

dim(Val(`n1)) = ?

H

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

dim(Val(`np)) = ∞

p 6= 1, 2:

• p 6= 1: geodesic ⇐⇒ convex

• p 6= 2: only isometries are translations, coordinate permutations, andreflections in coordinate hyperplanes

Rn with the p-norm

Write `np for Rn with metric d(x , y) =

(∑|xi − yi |p

)1/p(1 ≤ p < ∞).

1 2 p

groupisometrygeodesic

sets

1 2 p

groupisometrygeodesic

sets

dim(Val(`n2)) = n + 1

H

dim(Val(`n1)) = ?

H

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

dim(Val(`np)) = ∞

p 6= 1, 2:

• p 6= 1: geodesic ⇐⇒ convex

• p 6= 2: only isometries are translations, coordinate permutations, andreflections in coordinate hyperplanes

2. Convex geometry in `n1

Geodesic sets in `n1

A subset of Rn is `1-convex if it is geodesic in `n1.

The `1-convex sets include the convex sets—and much more.

Examples:

Geodesic sets in `n1

A subset of Rn is `1-convex if it is geodesic in `n1.

The `1-convex sets include the convex sets—and much more.

Examples:

Geodesic sets in `n1

A subset of Rn is `1-convex if it is geodesic in `n1.

The `1-convex sets include the convex sets—and much more.

Examples:

Geodesic sets in `n1

A subset of Rn is `1-convex if it is geodesic in `n1.

The `1-convex sets include the convex sets—and much more.

Examples:

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2

convex convex

`n1

`1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩

, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2

convex convex

`n1

`1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2

convex convex

`n1

`1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2

convex convex

`n1

`1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2

convex convex

`n1

`1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2

convex convex

`n1

`1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2

convex convex

`n1

`1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2

convex convex

`n1

`1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2 convex

convex

`n1

`1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2 convex convex

`n1

`1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2 convex convex

`n1 `1-convex

interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2 convex convex

`n1 `1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Why it’s not straightforward

The class of `1-convex sets is not closed under ∩, or +.

What’s going on?

Let A be a metric space. For a, a′ ∈ A, write

Γ(a, a′) = {distance-preserving paths from a to a′ in A}.

X ⊆ A is geodesic iff whenever x , x ′ ∈ X , ∃γ ∈ Γ(x , x ′) : image(γ) ⊆ X .

X ⊆ A is cogeodesic iff whenever x , x ′ ∈ X , ∀γ ∈ Γ(x , x ′), image(γ) ⊆ X .

geodesic cogeodesic

`n2 convex convex

`n1 `1-convex interval I1 × · · · × In

Sometimes the right generalization of ‘convex’ is ‘geodesic’.Sometimes it’s ‘cogeodesic’.

Some basic properties of `1-convex sets

•

Convex`1-convex

∩

convexinterval

=

convex`1-convex

•

Convex`1-convex

+

convexinterval

=

convex`1-convex

•

If X and Y are closed sets with X ∪ Y

convex,`1-convex,

and I is

convex,an interval,

then

(X ∩ Y ) + I = (X + I ) ∩ (Y + I ).

• If X and Y are closed sets then

X , Y , X ∪ Y `1-convex ⇒ X ∩ Y `1-convex.

Some basic properties of `1-convex sets

• Convex

`1-convex

∩ convex

interval

= convex

`1-convex

•

Convex`1-convex

+

convexinterval

=

convex`1-convex

•

If X and Y are closed sets with X ∪ Y

convex,`1-convex,

and I is

convex,an interval,

then

(X ∩ Y ) + I = (X + I ) ∩ (Y + I ).

• If X and Y are closed sets then

X , Y , X ∪ Y `1-convex ⇒ X ∩ Y `1-convex.

Some basic properties of `1-convex sets

•

Convex

`1-convex ∩

convex

interval =

convex

`1-convex

•

Convex`1-convex

+

convexinterval

=

convex`1-convex

•

If X and Y are closed sets with X ∪ Y

convex,`1-convex,

and I is

convex,an interval,

then

(X ∩ Y ) + I = (X + I ) ∩ (Y + I ).

• If X and Y are closed sets then

X , Y , X ∪ Y `1-convex ⇒ X ∩ Y `1-convex.

Some basic properties of `1-convex sets

•

Convex

`1-convex ∩

convex

interval =

convex

`1-convex

• Convex

`1-convex

+ convex

interval

= convex

`1-convex

•

If X and Y are closed sets with X ∪ Y

convex,`1-convex,

and I is

convex,an interval,

then

(X ∩ Y ) + I = (X + I ) ∩ (Y + I ).

• If X and Y are closed sets then

X , Y , X ∪ Y `1-convex ⇒ X ∩ Y `1-convex.

Some basic properties of `1-convex sets

•

Convex

`1-convex ∩

convex

interval =

convex

`1-convex

•

Convex

`1-convex +

convex

interval =

convex

`1-convex

•

If X and Y are closed sets with X ∪ Y

convex,`1-convex,

and I is

convex,an interval,

then

(X ∩ Y ) + I = (X + I ) ∩ (Y + I ).

• If X and Y are closed sets then

X , Y , X ∪ Y `1-convex ⇒ X ∩ Y `1-convex.

Some basic properties of `1-convex sets

•

Convex

`1-convex ∩

convex

interval =

convex

`1-convex

•

Convex

`1-convex +

convex

interval =

convex

`1-convex

•

If X and Y are closed sets with X ∪ Y

convex,`1-convex,

and I is

convex,an interval,

then

(X ∩ Y ) + I = (X + I ) ∩ (Y + I ).

• If X and Y are closed sets then

X , Y , X ∪ Y `1-convex ⇒ X ∩ Y `1-convex.

Some basic properties of `1-convex sets

•

Convex

`1-convex ∩

convex

interval =

convex

`1-convex

•

Convex

`1-convex +

convex

interval =

convex

`1-convex

• If X and Y are closed sets with X ∪ Y convex,

`1-convex,

and I is convex,

an interval,

then

(X ∩ Y ) + I = (X + I ) ∩ (Y + I ).

• If X and Y are closed sets then

X , Y , X ∪ Y `1-convex ⇒ X ∩ Y `1-convex.

Some basic properties of `1-convex sets

•

Convex

`1-convex ∩

convex

interval =

convex

`1-convex

•

Convex

`1-convex +

convex

interval =

convex

`1-convex

• If X and Y are closed sets with X ∪ Y

convex,

`1-convex,and I is

convex,

an interval, then

(X ∩ Y ) + I = (X + I ) ∩ (Y + I ).

• If X and Y are closed sets then

X , Y , X ∪ Y `1-convex ⇒ X ∩ Y `1-convex.

Some basic properties of `1-convex sets

•

Convex

`1-convex ∩

convex

interval =

convex

`1-convex

•

Convex

`1-convex +

convex

interval =

convex

`1-convex

• If X and Y are closed sets with X ∪ Y

convex,

`1-convex,and I is

convex,

an interval, then

(X ∩ Y ) + I = (X + I ) ∩ (Y + I ).

• If X and Y are closed sets then

X , Y , X ∪ Y `1-convex ⇒ X ∩ Y `1-convex.

3. Integral geometry in `n1

The `1-intrinsic volumes

A coordinate subspace of Rn is a subspace spanned by some subset of thestandard basis.

For 0 ≤ i ≤ n, write

Gr′n,i = {i-dimensional coordinate subspaces of Rn}

(a finite set).

For P ∈ Gr′n,i , write orthogonal projection Rn → P as X 7→ X |P.

The

ith intrinsic volumeith `1-intrinsic volume

of a compact

convex`1-convex

set X is

Vi (X ) = const ·∫

Grn,i

Voli (X |P) dP.V ′i (X ) =

∑P∈Gr′n,i

Voli (X |P).

It is a continuous invariant valuation on `1-convex sets,homogeneous of degree i .

The `1-intrinsic volumes

A coordinate subspace of Rn is a subspace spanned by some subset of thestandard basis.

For 0 ≤ i ≤ n, write

Gr′n,i = {i-dimensional coordinate subspaces of Rn}

(a finite set).

For P ∈ Gr′n,i , write orthogonal projection Rn → P as X 7→ X |P.

The

ith intrinsic volumeith `1-intrinsic volume

of a compact

convex`1-convex

set X is

Vi (X ) = const ·∫

Grn,i

Voli (X |P) dP.V ′i (X ) =

∑P∈Gr′n,i

Voli (X |P).

It is a continuous invariant valuation on `1-convex sets,homogeneous of degree i .

The `1-intrinsic volumes

A coordinate subspace of Rn is a subspace spanned by some subset of thestandard basis.

For 0 ≤ i ≤ n, write

Gr′n,i = {i-dimensional coordinate subspaces of Rn}

(a finite set).

For P ∈ Gr′n,i , write orthogonal projection Rn → P as X 7→ X |P.

The

ith intrinsic volumeith `1-intrinsic volume

of a compact

convex`1-convex

set X is

Vi (X ) = const ·∫

Grn,i

Voli (X |P) dP.V ′i (X ) =

∑P∈Gr′n,i

Voli (X |P).

It is a continuous invariant valuation on `1-convex sets,homogeneous of degree i .

The `1-intrinsic volumes

A coordinate subspace of Rn is a subspace spanned by some subset of thestandard basis.

For 0 ≤ i ≤ n, write

Gr′n,i = {i-dimensional coordinate subspaces of Rn}

(a finite set).

For P ∈ Gr′n,i , write orthogonal projection Rn → P as X 7→ X |P.

The

ith intrinsic volumeith `1-intrinsic volume

of a compact

convex`1-convex

set X is

Vi (X ) = const ·∫

Grn,i

Voli (X |P) dP.V ′i (X ) =

∑P∈Gr′n,i

Voli (X |P).

It is a continuous invariant valuation on `1-convex sets,homogeneous of degree i .

The `1-intrinsic volumes

A coordinate subspace of Rn is a subspace spanned by some subset of thestandard basis.

For 0 ≤ i ≤ n, write

Gr′n,i = {i-dimensional coordinate subspaces of Rn}

(a finite set).

For P ∈ Gr′n,i , write orthogonal projection Rn → P as X 7→ X |P.

The ith intrinsic volume

ith `1-intrinsic volume

of a compact convex

`1-convex

set X is

Vi (X ) = const ·∫

Grn,i

Voli (X |P) dP.

V ′i (X ) =

∑P∈Gr′n,i

Voli (X |P).

It is a continuous invariant valuation on `1-convex sets,homogeneous of degree i .

The `1-intrinsic volumes

A coordinate subspace of Rn is a subspace spanned by some subset of thestandard basis.

For 0 ≤ i ≤ n, write

Gr′n,i = {i-dimensional coordinate subspaces of Rn}

(a finite set).

For P ∈ Gr′n,i , write orthogonal projection Rn → P as X 7→ X |P.

The

ith intrinsic volume

ith `1-intrinsic volume of a compact

convex

`1-convex set X is

Vi (X ) = const ·∫

Grn,i

Voli (X |P) dP.

V ′i (X ) =

∑P∈Gr′n,i

Voli (X |P).

It is a continuous invariant valuation on `1-convex sets,homogeneous of degree i .

The `1-intrinsic volumes

A coordinate subspace of Rn is a subspace spanned by some subset of thestandard basis.

For 0 ≤ i ≤ n, write

Gr′n,i = {i-dimensional coordinate subspaces of Rn}

(a finite set).

For P ∈ Gr′n,i , write orthogonal projection Rn → P as X 7→ X |P.

The

ith intrinsic volume

ith `1-intrinsic volume of a compact

convex

`1-convex set X is

Vi (X ) = const ·∫

Grn,i

Voli (X |P) dP.

V ′i (X ) =

∑P∈Gr′n,i

Voli (X |P).

It is a continuous invariant valuation on `1-convex sets

,homogeneous of degree i .

The `1-intrinsic volumes

A coordinate subspace of Rn is a subspace spanned by some subset of thestandard basis.

For 0 ≤ i ≤ n, write

Gr′n,i = {i-dimensional coordinate subspaces of Rn}

(a finite set).

For P ∈ Gr′n,i , write orthogonal projection Rn → P as X 7→ X |P.

The

ith intrinsic volume

ith `1-intrinsic volume of a compact

convex

`1-convex set X is

Vi (X ) = const ·∫

Grn,i

Voli (X |P) dP.

V ′i (X ) =

∑P∈Gr′n,i

Voli (X |P).

It is a continuous invariant valuation on `1-convex sets,homogeneous of degree i .

A Hadwiger-type theorem

Theorem

The `1-intrinsic volumes V ′0, . . . ,V

′n form a basis for the vector space Val(`n

1)of continuous invariant valuations on `1-convex subsets of Rn.

In particular, dim(Val(`n1)) = n + 1.

The proof relies heavily on specific features of the geometry of `n1.

A Hadwiger-type theorem

Theorem

The `1-intrinsic volumes V ′0, . . . ,V

′n form a basis for the vector space Val(`n

1)of continuous invariant valuations on `1-convex subsets of Rn.

In particular, dim(Val(`n1)) = n + 1.

The proof relies heavily on specific features of the geometry of `n1.

A Hadwiger-type theorem

Theorem

The `1-intrinsic volumes V ′0, . . . ,V

′n form a basis for the vector space Val(`n

1)of continuous invariant valuations on `1-convex subsets of Rn.

In particular, dim(Val(`n1)) = n + 1.

The proof relies heavily on specific features of the geometry of `n1.

A Hadwiger-type theorem

Theorem

The `1-intrinsic volumes V ′0, . . . ,V

′n form a basis for the vector space Val(`n

1)of continuous invariant valuations on `1-convex subsets of Rn.

In particular, dim(Val(`n1)) = n + 1.

The proof relies heavily on specific features of the geometry of `n1.

A Steiner-type formula

Theorem

Let X ⊆ Rn be a compact `1-convex set. Let λ ≥ 0. Then

Vol(X + λ[−1

2 , 12 ]n

)=

n∑i=0

V ′i (X )λn−i .

This uses the unit cube (a ball in `n∞), not the unit ball in `n

1.

There is no Steiner-type formula using the ball.

A Steiner-type formula

Theorem

Let X ⊆ Rn be a compact `1-convex set. Let λ ≥ 0. Then

Vol(X + λ[−1

2 , 12 ]n

)=

n∑i=0

V ′i (X )λn−i .

This uses the unit cube (a ball in `n∞), not the unit ball in `n

1.

There is no Steiner-type formula using the ball.

A Steiner-type formula

Theorem

Let X ⊆ Rn be a compact `1-convex set. Let λ ≥ 0. Then

Vol(X + λ[−1

2 , 12 ]n

)=

n∑i=0

V ′i (X )λn−i .

This uses the unit cube (a ball in `n∞), not the unit ball in `n

1.

There is no Steiner-type formula using the ball.

A Steiner-type formula

Theorem

Let X ⊆ Rn be a compact `1-convex set. Let λ ≥ 0. Then

Vol(X + λ[−1

2 , 12 ]n

)=

n∑i=0

V ′i (X )λn−i .

This uses the unit cube (a ball in `n∞), not the unit ball in `n

1.

There is no Steiner-type formula using the ball.

A Crofton-type formula

For 0 ≤ k ≤ n, write

Graff ′n,k ={k-dimensional affine subspaces of Rn

parallel to some coordinate subspace}.

It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set. Let 0 ≤ k ≤ n. Then

measure({A ∈ Graff ′n,k | X ∩ A 6= ∅}

)= V ′

n−k(X ).

More generally, for 0 ≤ j ≤ k ≤ n,∫Graff′

n,k

V ′j (X ∩ A) dA =

(n + j − k

j

)V ′

n+j−k(X ).

A Crofton-type formula

For 0 ≤ k ≤ n, write

Graff ′n,k ={k-dimensional affine subspaces of Rn

parallel to some coordinate subspace}.

It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set. Let 0 ≤ k ≤ n. Then

measure({A ∈ Graff ′n,k | X ∩ A 6= ∅}

)= V ′

n−k(X ).

More generally, for 0 ≤ j ≤ k ≤ n,∫Graff′

n,k

V ′j (X ∩ A) dA =

(n + j − k

j

)V ′

n+j−k(X ).

A Crofton-type formula

For 0 ≤ k ≤ n, write

Graff ′n,k ={k-dimensional affine subspaces of Rn

parallel to some coordinate subspace}.

It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set. Let 0 ≤ k ≤ n. Then

measure({A ∈ Graff ′n,k | X ∩ A 6= ∅}

)= V ′

n−k(X ).

More generally, for 0 ≤ j ≤ k ≤ n,∫Graff′

n,k

V ′j (X ∩ A) dA =

(n + j − k

j

)V ′

n+j−k(X ).

A Crofton-type formula

For 0 ≤ k ≤ n, write

Graff ′n,k ={k-dimensional affine subspaces of Rn

parallel to some coordinate subspace}.

It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set. Let 0 ≤ k ≤ n.

Then

measure({A ∈ Graff ′n,k | X ∩ A 6= ∅}

)= V ′

n−k(X ).

More generally, for 0 ≤ j ≤ k ≤ n,∫Graff′

n,k

V ′j (X ∩ A) dA =

(n + j − k

j

)V ′

n+j−k(X ).

A Crofton-type formula

For 0 ≤ k ≤ n, write

Graff ′n,k ={k-dimensional affine subspaces of Rn

parallel to some coordinate subspace}.

It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set. Let 0 ≤ k ≤ n. Then

measure({A ∈ Graff ′n,k | X ∩ A 6= ∅}

)= V ′

n−k(X ).

More generally, for 0 ≤ j ≤ k ≤ n,∫Graff′

n,k

V ′j (X ∩ A) dA =

(n + j − k

j

)V ′

n+j−k(X ).

A Crofton-type formula

For 0 ≤ k ≤ n, write

Graff ′n,k ={k-dimensional affine subspaces of Rn

parallel to some coordinate subspace}.

It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set. Let 0 ≤ k ≤ n. Then

measure({A ∈ Graff ′n,k | X ∩ A 6= ∅}

)= V ′

n−k(X ).

More generally, for 0 ≤ j ≤ k ≤ n,∫Graff′

n,k

V ′j (X ∩ A) dA =

(n + j − k

j

)V ′

n+j−k(X ).

Kinematic formulas

Write Gn for the isometry group of `n1. It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set, and I ⊆ Rn a compact interval.Then

measure({g ∈ Gn | gX ∩ I 6= ∅}

)=

∑i+j=n

(n

i

)−1

V ′i (X )V ′

j (I ).

More generally, for 0 ≤ k ≤ n,∫Gn

V ′k(gX ∩ I ) dg =

∑i+j=n+k

(n

i

)−1( j

k

)V ′

i (X )V ′j (I ).

The constants in the Crofton and kinematic formulas are the same as in theclassical case, except that the flag coefficients

[mr

]are replaced by the

binomial coefficients(m

r

).

Kinematic formulasWrite Gn for the isometry group of `n

1.

It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set, and I ⊆ Rn a compact interval.Then

measure({g ∈ Gn | gX ∩ I 6= ∅}

)=

∑i+j=n

(n

i

)−1

V ′i (X )V ′

j (I ).

More generally, for 0 ≤ k ≤ n,∫Gn

V ′k(gX ∩ I ) dg =

∑i+j=n+k

(n

i

)−1( j

k

)V ′

i (X )V ′j (I ).

The constants in the Crofton and kinematic formulas are the same as in theclassical case, except that the flag coefficients

[mr

]are replaced by the

binomial coefficients(m

r

).

Kinematic formulasWrite Gn for the isometry group of `n

1. It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set, and I ⊆ Rn a compact interval.Then

measure({g ∈ Gn | gX ∩ I 6= ∅}

)=

∑i+j=n

(n

i

)−1

V ′i (X )V ′

j (I ).

More generally, for 0 ≤ k ≤ n,∫Gn

V ′k(gX ∩ I ) dg =

∑i+j=n+k

(n

i

)−1( j

k

)V ′

i (X )V ′j (I ).

The constants in the Crofton and kinematic formulas are the same as in theclassical case, except that the flag coefficients

[mr

]are replaced by the

binomial coefficients(m

r

).

Kinematic formulasWrite Gn for the isometry group of `n

1. It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set, and I ⊆ Rn a compact interval.

Then

measure({g ∈ Gn | gX ∩ I 6= ∅}

)=

∑i+j=n

(n

i

)−1

V ′i (X )V ′

j (I ).

More generally, for 0 ≤ k ≤ n,∫Gn

V ′k(gX ∩ I ) dg =

∑i+j=n+k

(n

i

)−1( j

k

)V ′

i (X )V ′j (I ).

The constants in the Crofton and kinematic formulas are the same as in theclassical case, except that the flag coefficients

[mr

]are replaced by the

binomial coefficients(m

r

).

Kinematic formulasWrite Gn for the isometry group of `n

1. It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set, and I ⊆ Rn a compact interval.Then

measure({g ∈ Gn | gX ∩ I 6= ∅}

)=

∑i+j=n

(n

i

)−1

V ′i (X )V ′

j (I ).

More generally, for 0 ≤ k ≤ n,∫Gn

V ′k(gX ∩ I ) dg =

∑i+j=n+k

(n

i

)−1( j

k

)V ′

i (X )V ′j (I ).

The constants in the Crofton and kinematic formulas are the same as in theclassical case, except that the flag coefficients

[mr

]are replaced by the

binomial coefficients(m

r

).

Kinematic formulasWrite Gn for the isometry group of `n

1. It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set, and I ⊆ Rn a compact interval.Then

measure({g ∈ Gn | gX ∩ I 6= ∅}

)=

∑i+j=n

(n

i

)−1

V ′i (X )V ′

j (I ).

More generally, for 0 ≤ k ≤ n,∫Gn

V ′k(gX ∩ I ) dg =

∑i+j=n+k

(n

i

)−1( j

k

)V ′

i (X )V ′j (I ).

The constants in the Crofton and kinematic formulas are the same as in theclassical case, except that the flag coefficients

[mr

]are replaced by the

binomial coefficients(m

r

).

Kinematic formulasWrite Gn for the isometry group of `n

1. It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set, and I ⊆ Rn a compact interval.Then

measure({g ∈ Gn | gX ∩ I 6= ∅}

)=

∑i+j=n

(n

i

)−1

V ′i (X )V ′

j (I ).

More generally, for 0 ≤ k ≤ n,∫Gn

V ′k(gX ∩ I ) dg =

∑i+j=n+k

(n

i

)−1( j

k

)V ′

i (X )V ′j (I ).

The constants in the Crofton and kinematic formulas are the same as in theclassical case

, except that the flag coefficients[mr

]are replaced by the

binomial coefficients(m

r

).

Kinematic formulasWrite Gn for the isometry group of `n

1. It carries a natural invariant measure.

Theorem

Let X ⊆ Rn be a compact `1-convex set, and I ⊆ Rn a compact interval.Then

measure({g ∈ Gn | gX ∩ I 6= ∅}

)=

∑i+j=n

(n

i

)−1

V ′i (X )V ′

j (I ).

More generally, for 0 ≤ k ≤ n,∫Gn

V ′k(gX ∩ I ) dg =

∑i+j=n+k

(n

i

)−1( j

k

)V ′

i (X )V ′j (I ).

The constants in the Crofton and kinematic formulas are the same as in theclassical case, except that the flag coefficients

[mr

]are replaced by the

binomial coefficients(m

r

).

Digression: magnitude

There is a single category-theoretic definition that producesinvariants of ‘size’ in several branches of mathematics.

E.g. in set theory: cardinality. In topology: Euler characteristic.

For metric spaces, it produces a real-valued invariant called magnitude.

For compact subsets X of `n1 or `n

2, it can be defined (M. Meckes) by

|X | = sup

{µ(X )2∫

X

∫X e−d(x ,y) dµ(x) dµ(y)

∣∣∣∣ signed Borel measures µ on X

}.

Conjecture()

For

compact convexcompact `1-convexconvex bodies X ⊆

`n2

,

|X | =n∑

i=0

1

i ! Vol(B i )

ViV ′i

(X ).

Digression: magnitude

There is a single category-theoretic definition that producesinvariants of ‘size’ in several branches of mathematics.

E.g. in set theory: cardinality. In topology: Euler characteristic.

For metric spaces, it produces a real-valued invariant called magnitude.

For compact subsets X of `n1 or `n

2, it can be defined (M. Meckes) by

|X | = sup

{µ(X )2∫

X

∫X e−d(x ,y) dµ(x) dµ(y)

∣∣∣∣ signed Borel measures µ on X

}.

Conjecture()

For

compact convexcompact `1-convexconvex bodies X ⊆

`n2

,

|X | =n∑

i=0

1

i ! Vol(B i )

ViV ′i

(X ).

Digression: magnitude

There is a single category-theoretic definition that producesinvariants of ‘size’ in several branches of mathematics.

E.g. in set theory: cardinality. In topology: Euler characteristic.

For metric spaces, it produces a real-valued invariant called magnitude.

For compact subsets X of `n1 or `n

2, it can be defined (M. Meckes) by

|X | = sup

{µ(X )2∫

X

∫X e−d(x ,y) dµ(x) dµ(y)

∣∣∣∣ signed Borel measures µ on X

}.

Conjecture()

For

compact convexcompact `1-convexconvex bodies X ⊆

`n2

,

|X | =n∑

i=0

1

i ! Vol(B i )

ViV ′i

(X ).

Digression: magnitude

There is a single category-theoretic definition that producesinvariants of ‘size’ in several branches of mathematics.

E.g. in set theory: cardinality. In topology: Euler characteristic.

For metric spaces, it produces a real-valued invariant called magnitude.

For compact subsets X of `n1 or `n

2, it can be defined (M. Meckes) by

|X | = sup

{µ(X )2∫

X

∫X e−d(x ,y) dµ(x) dµ(y)

∣∣∣∣ signed Borel measures µ on X

}.

Conjecture()

For

compact convexcompact `1-convexconvex bodies X ⊆

`n2

,

|X | =n∑

i=0

1

i ! Vol(B i )

ViV ′i

(X ).

Digression: magnitude

There is a single category-theoretic definition that producesinvariants of ‘size’ in several branches of mathematics.

E.g. in set theory: cardinality. In topology: Euler characteristic.

For metric spaces, it produces a real-valued invariant called magnitude.

For compact subsets X of `n1 or `n

2, it can be defined (M. Meckes) by

|X | = sup

{µ(X )2∫

X

∫X e−d(x ,y) dµ(x) dµ(y)

∣∣∣∣ signed Borel measures µ on X

}.

Conjecture()

For

compact convexcompact `1-convexconvex bodies X ⊆

`n2

,

|X | =n∑

i=0

1

i ! Vol(B i )

ViV ′i

(X ).

Digression: magnitude

There is a single category-theoretic definition that producesinvariants of ‘size’ in several branches of mathematics.

E.g. in set theory: cardinality. In topology: Euler characteristic.

For metric spaces, it produces a real-valued invariant called magnitude.

For compact subsets X of `n1 or `n

2, it can be defined (M. Meckes) by

|X | = sup

{µ(X )2∫

X

∫X e−d(x ,y) dµ(x) dµ(y)

∣∣∣∣ signed Borel measures µ on X

}.

Conjecture (with S. Willerton)

For

compact convexcompact `1-convexconvex bodies X ⊆

`n2

,

|X | =n∑

i=0

1

i ! Vol(B i )

ViV ′i

(X ).

Digression: magnitude

There is a single category-theoretic definition that producesinvariants of ‘size’ in several branches of mathematics.

E.g. in set theory: cardinality. In topology: Euler characteristic.

For metric spaces, it produces a real-valued invariant called magnitude.

For compact subsets X of `n1 or `n

2, it can be defined (M. Meckes) by

|X | = sup

{µ(X )2∫

X

∫X e−d(x ,y) dµ(x) dµ(y)

∣∣∣∣ signed Borel measures µ on X

}.

Conjecture (with S. Willerton)

For compact convex

compact `1-convexconvex bodies

X ⊆ `n2,

|X | =

n∑i=0

1

i ! Vol(B i )

ViV ′i

(X ).

Digression: magnitude

There is a single category-theoretic definition that producesinvariants of ‘size’ in several branches of mathematics.

E.g. in set theory: cardinality. In topology: Euler characteristic.

For metric spaces, it produces a real-valued invariant called magnitude.

For compact subsets X of `n1 or `n

2, it can be defined (M. Meckes) by

|X | = sup

{µ(X )2∫

X

∫X e−d(x ,y) dµ(x) dµ(y)

∣∣∣∣ signed Borel measures µ on X

}.

Conjecture (with S. Willerton)

For compact convex

compact `1-convexconvex bodies

X ⊆ `n2,

|X | =n∑

i=0

1

i ! Vol(B i )Vi

V ′i

(X ).

Digression: magnitude

There is a single category-theoretic definition that producesinvariants of ‘size’ in several branches of mathematics.

E.g. in set theory: cardinality. In topology: Euler characteristic.

For metric spaces, it produces a real-valued invariant called magnitude.

For compact subsets X of `n1 or `n

2, it can be defined (M. Meckes) by

|X | = sup

{µ(X )2∫

X

∫X e−d(x ,y) dµ(x) dµ(y)

∣∣∣∣ signed Borel measures µ on X

}.

Conjecture()

For

compact convex

compact `1-convex

convex bodies

X ⊆ `n1,

|X | =n∑

i=0

1

i ! Vol(B i )

Vi

V ′i (X ).

Digression: magnitude

There is a single category-theoretic definition that producesinvariants of ‘size’ in several branches of mathematics.

E.g. in set theory: cardinality. In topology: Euler characteristic.

For metric spaces, it produces a real-valued invariant called magnitude.

For compact subsets X of `n1 or `n

2, it can be defined (M. Meckes) by

|X | = sup

{µ(X )2∫

X

∫X e−d(x ,y) dµ(x) dµ(y)

∣∣∣∣ signed Borel measures µ on X

}.

Theoremj()

For

compact convexcompact `1-convex

convex bodies X ⊆ `n1,

|X | =n∑

i=0

1

i ! Vol(B i )

Vi

V ′i (X ).

Summary

Summary

1 2 p

groupisometrygeodesic

sets

p = 1, 2: very similar to each otherHH

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

p 6= 1, 2: different

`n1 and `n

2 have strikingly similar integral geometry, even though:

• they have different geodesic subsets

• they have different isometry groups

• the proofs of the theorems are quite different.

Summary

1 2 p

groupisometrygeodesic

sets

p = 1, 2: very similar to each otherHH

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

p 6= 1, 2: different

`n1 and `n

2 have strikingly similar integral geometry, even though:

• they have different geodesic subsets

• they have different isometry groups

• the proofs of the theorems are quite different.

Summary

1 2 p

groupisometrygeodesic

sets

p = 1, 2: very similar to each otherHH

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

p 6= 1, 2: different

`n1 and `n

2 have strikingly similar integral geometry, even though:

• they have different geodesic subsets

• they have different isometry groups

• the proofs of the theorems are quite different.

Summary

1 2 p

groupisometrygeodesic

sets

p = 1, 2: very similar to each otherHH

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

p 6= 1, 2: different

`n1 and `n

2 have strikingly similar integral geometry, even though:

• they have different geodesic subsets

• they have different isometry groups

• the proofs of the theorems are quite different.

Summary

1 2 p

groupisometrygeodesic

sets

p = 1, 2: very similar to each otherHH

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

p 6= 1, 2: different

`n1 and `n

2 have strikingly similar integral geometry, even though:

• they have different geodesic subsets

• they have different isometry groups

• the proofs of the theorems are quite different.

Summary

1 2 p

groupisometrygeodesic

sets

p = 1, 2: very similar to each otherHH

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

p 6= 1, 2: different

`n1 and `n

2 have strikingly similar integral geometry, even though:

• they have different geodesic subsets

• they have different isometry groups

• the proofs of the theorems are quite different.

Summary

1 2 p

groupisometrygeodesic

sets

p = 1, 2: very similar to each otherHH

︸ ︷︷ ︸ ︸ ︷︷ ︸N N

p 6= 1, 2: different

`n1 and `n

2 have strikingly similar integral geometry, even though:

• they have different geodesic subsets

• they have different isometry groups

• the proofs of the theorems are quite different.

References

• Tom Leinster, Integral geometry for the 1-norm, arXiv:1012.5881v2.

• Tom Leinster, The magnitude of metric spaces, arXiv:1012.5857.

• Tom Leinster, Simon Willerton, On the asymptotic magnitude ofsubsets of Euclidean space, arXiv:0908.1582.

• Mark Meckes, Positive definite metric spaces, arXiv:1012.5863.