Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals

Mohan Ravichandran, Bogazici, Istanbul

IML, Unimodality, Log Concavity and Beyond March 19, 2020

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 1 / 28 Table of contents

1 Valuations

2 Ehrhart Positivity

3 Generalized Permutohedra

4 Open Problems

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 2 / 28 A map φ : Kd → R is called a valuation if for every A, B ∈ Kd , φ(A ∪ B) = φ(A) + φ(B) − φ(A ∩ B),

whenever A ∪ B is also in Kd . Some examples of valuations : 1 Volume. 2 Surface area. 3 Volume of slice with a ﬁxed hyperplane. 4 Expected Volume of intersection with random k dimensional subspace (uniformly distributed with respect to the unique O(d) invariant measure on G(d, k)).

Let us call the valuations in the last item Vk for 0 ≤ k ≤ d.

Note that V0 = 1 and Vd−1 is the surface area and Vd is the usual volume.

V1 is the mean width. One reason for studying valuations is Hilbert’s third problem.

Valuations Valuations on Convex Sets

d Kd : The class of convex sets in R .

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 3 / 28 Some examples of valuations : 1 Volume. 2 Surface area. 3 Volume of slice with a ﬁxed hyperplane. 4 Expected Volume of intersection with random k dimensional subspace (uniformly distributed with respect to the unique O(d) invariant measure on G(d, k)).

Let us call the valuations in the last item Vk for 0 ≤ k ≤ d.

Note that V0 = 1 and Vd−1 is the surface area and Vd is the usual volume.

V1 is the mean width. One reason for studying valuations is Hilbert’s third problem.

Valuations Valuations on Convex Sets

d Kd : The class of convex sets in R .

A map φ : Kd → R is called a valuation if for every A, B ∈ Kd , φ(A ∪ B) = φ(A) + φ(B) − φ(A ∩ B), whenever A ∪ B is also in Kd .

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 3 / 28 Let us call the valuations in the last item Vk for 0 ≤ k ≤ d.

Note that V0 = 1 and Vd−1 is the surface area and Vd is the usual volume.

V1 is the mean width. One reason for studying valuations is Hilbert’s third problem.

Valuations Valuations on Convex Sets

d Kd : The class of convex sets in R .

A map φ : Kd → R is called a valuation if for every A, B ∈ Kd , φ(A ∪ B) = φ(A) + φ(B) − φ(A ∩ B), whenever A ∪ B is also in Kd . Some examples of valuations : 1 Volume. 2 Surface area. 3 Volume of slice with a ﬁxed hyperplane. 4 Expected Volume of intersection with random k dimensional subspace (uniformly distributed with respect to the unique O(d) invariant measure on G(d, k)).

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 3 / 28 V1 is the mean width. One reason for studying valuations is Hilbert’s third problem.

Valuations Valuations on Convex Sets

d Kd : The class of convex sets in R .

Let us call the valuations in the last item Vk for 0 ≤ k ≤ d.

Note that V0 = 1 and Vd−1 is the surface area and Vd is the usual volume.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 3 / 28 One reason for studying valuations is Hilbert’s third problem.

Valuations Valuations on Convex Sets

d Kd : The class of convex sets in R .

Let us call the valuations in the last item Vk for 0 ≤ k ≤ d.

Note that V0 = 1 and Vd−1 is the surface area and Vd is the usual volume.

V1 is the mean width.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 3 / 28 One reason for studying valuations is Hilbert’s third problem.

Valuations Valuations on Convex Sets

d Kd : The class of convex sets in R .

Let us call the valuations in the last item Vk for 0 ≤ k ≤ d.

Note that V0 = 1 and Vd−1 is the surface area and Vd is the usual volume.

V1 is the mean width.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 3 / 28 Valuations Valuations on Convex Sets

d Kd : The class of convex sets in R .

Let us call the valuations in the last item Vk for 0 ≤ k ≤ d.

Note that V0 = 1 and Vd−1 is the surface area and Vd is the usual volume.

V1 is the mean width. One reason for studying valuations is Hilbert’s third problem.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 3 / 28 2 k Two polytopes A, B ⊂ R are called D2 equidissectable if there are dissections A = ∪i=1Ai and n B = ∪i=1Bi and such that Ai ∼D2 Bi for each i. Caveat : What we are calling equidissectability is usually called Scissors Congruence.

The most elementary way of computing volume is to break down a set into elementary sets and sum up their elementary volumes. To this end, we have

Theorem (Bolyai-Gerwein)

Two plane polygons are of equal area iﬀ they are D2 equidissectable. Hilbert’s third problem (essentially) asked the following

Question (Hilbert’s third problem) Does the Bolyai-Gerwein result extend to higher dimensions?

This was the ﬁrst of Hilbert’s 23 problems to be solved, actually in the same year!

Valuations Hilbert’s third problem

2 Let D2 be the group of isometries of R . Deﬁnition 2 k A dissection of a polytope A ⊂ R is a collection of polytopes A1,..., Ak such that A = ∪i=1Ai and such that the interiors of the Ai are disjoint.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 4 / 28 The most elementary way of computing volume is to break down a set into elementary sets and sum up their elementary volumes. To this end, we have

Theorem (Bolyai-Gerwein)

Two plane polygons are of equal area iﬀ they are D2 equidissectable. Hilbert’s third problem (essentially) asked the following

Question (Hilbert’s third problem) Does the Bolyai-Gerwein result extend to higher dimensions?

This was the ﬁrst of Hilbert’s 23 problems to be solved, actually in the same year!

Valuations Hilbert’s third problem

2 k Two polytopes A, B ⊂ R are called D2 equidissectable if there are dissections A = ∪i=1Ai and n B = ∪i=1Bi and such that Ai ∼D2 Bi for each i. Caveat : What we are calling equidissectability is usually called Scissors Congruence.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 4 / 28 Hilbert’s third problem (essentially) asked the following

Question (Hilbert’s third problem) Does the Bolyai-Gerwein result extend to higher dimensions?

This was the ﬁrst of Hilbert’s 23 problems to be solved, actually in the same year!

Valuations Hilbert’s third problem

The most elementary way of computing volume is to break down a set into elementary sets and sum up their elementary volumes. To this end, we have

Theorem (Bolyai-Gerwein)

Two plane polygons are of equal area iﬀ they are D2 equidissectable.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 4 / 28 This was the ﬁrst of Hilbert’s 23 problems to be solved, actually in the same year!

Valuations Hilbert’s third problem

The most elementary way of computing volume is to break down a set into elementary sets and sum up their elementary volumes. To this end, we have

Theorem (Bolyai-Gerwein)

Two plane polygons are of equal area iﬀ they are D2 equidissectable. Hilbert’s third problem (essentially) asked the following

Question (Hilbert’s third problem) Does the Bolyai-Gerwein result extend to higher dimensions?

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 4 / 28 Valuations Hilbert’s third problem

The most elementary way of computing volume is to break down a set into elementary sets and sum up their elementary volumes. To this end, we have

Theorem (Bolyai-Gerwein)

Two plane polygons are of equal area iﬀ they are D2 equidissectable. Hilbert’s third problem (essentially) asked the following

Question (Hilbert’s third problem) Does the Bolyai-Gerwein result extend to higher dimensions?

This was the ﬁrst of Hilbert’s 23 problems to be solved, actually in the same year!

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 4 / 28 Dehn showed the necessity of the following (settling Hilbert’s third problem). The suﬃciency was settled by Sydler in 1965.

Theorem (Dehn, Sydler) 3 Polytopes P, Q ⊂ R are equidissectable under D3 iﬀ all their Dehn invariants are the same. It is easy to see that every Dehn invariant of the standard cube is zero while every Dehn invariant of the regular tetrahedron is non-zero.

There is a great back story to this : This, unknown to Hilbert had been proposed by Wladyslaw Kretkowski in 1882 for a prize contest in Krakow and solved by Ludwig Birkenmajer in 1884. "Equidecomposability of Polyhedra: A Solution of Hilbert’s Third Problem in Kraków before ICM 1900", Mathematical Intelligencer, https://link.springer.com/article/10.1007

Valuations Dehn Invariants

Deﬁnition Let f : R → R be an additive map that is zero at π but is not identically zero. Such a f must be 3 non-measurable. The associated Dehn invariant for a polytope P ⊂ R is

k ∗ X f (P) = σi f (αi ), i=1 where the σi are the edge lengths and the αi are the dihedral angles.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 5 / 28 There is a great back story to this : This, unknown to Hilbert had been proposed by Wladyslaw Kretkowski in 1882 for a prize contest in Krakow and solved by Ludwig Birkenmajer in 1884. "Equidecomposability of Polyhedra: A Solution of Hilbert’s Third Problem in Kraków before ICM 1900", Mathematical Intelligencer, https://link.springer.com/article/10.1007

Valuations Dehn Invariants

Deﬁnition Let f : R → R be an additive map that is zero at π but is not identically zero. Such a f must be 3 non-measurable. The associated Dehn invariant for a polytope P ⊂ R is

k ∗ X f (P) = σi f (αi ), i=1 where the σi are the edge lengths and the αi are the dihedral angles. Dehn showed the necessity of the following (settling Hilbert’s third problem). The suﬃciency was settled by Sydler in 1965.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 5 / 28 "Equidecomposability of Polyhedra: A Solution of Hilbert’s Third Problem in Kraków before ICM 1900", Mathematical Intelligencer, https://link.springer.com/article/10.1007

Valuations Dehn Invariants

Deﬁnition Let f : R → R be an additive map that is zero at π but is not identically zero. Such a f must be 3 non-measurable. The associated Dehn invariant for a polytope P ⊂ R is

There is a great back story to this : This, unknown to Hilbert had been proposed by Wladyslaw Kretkowski in 1882 for a prize contest in Krakow and solved by Ludwig Birkenmajer in 1884.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 5 / 28 Valuations Dehn Invariants

Deﬁnition Let f : R → R be an additive map that is zero at π but is not identically zero. Such a f must be 3 non-measurable. The associated Dehn invariant for a polytope P ⊂ R is

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 5 / 28 There are many problems related to Hilbert’s third problem that are still open. The most important one is :

Question Does the Dehn-Sydler theorem hold in spherical or hyperbolic geometry?

Some related results : Two sets are equidecomposable (as opposed to equidissectable) if they can be partitioned into D2 congruent sets.

Theorem (Laczkovich) Two plane polygons of equal area are equidecomposable : Indeed, one only needs to take translations in place of D2. A circular disk and a square of the same area are equidecomposable, even under the group of translations. And of course, we have the Banach-Tarski paradox.

Theorem n Any two bounded sets X , Y ⊂ R with non-empty interior, for n ≥ 3 are equidecomposable.

Valuations Valuations

3 What Dehn showed is: There are valuations on polytopes in R other than the volume.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 6 / 28 Some related results : Two sets are equidecomposable (as opposed to equidissectable) if they can be partitioned into D2 congruent sets.

Theorem n Any two bounded sets X , Y ⊂ R with non-empty interior, for n ≥ 3 are equidecomposable.

Valuations Valuations

3 What Dehn showed is: There are valuations on polytopes in R other than the volume. There are many problems related to Hilbert’s third problem that are still open. The most important one is :

Question Does the Dehn-Sydler theorem hold in spherical or hyperbolic geometry?

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 6 / 28 A circular disk and a square of the same area are equidecomposable, even under the group of translations. And of course, we have the Banach-Tarski paradox.

Theorem n Any two bounded sets X , Y ⊂ R with non-empty interior, for n ≥ 3 are equidecomposable.

Valuations Valuations

3 What Dehn showed is: There are valuations on polytopes in R other than the volume. There are many problems related to Hilbert’s third problem that are still open. The most important one is :

Question Does the Dehn-Sydler theorem hold in spherical or hyperbolic geometry?

Some related results : Two sets are equidecomposable (as opposed to equidissectable) if they can be partitioned into D2 congruent sets.

Theorem (Laczkovich) Two plane polygons of equal area are equidecomposable : Indeed, one only needs to take translations in place of D2.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 6 / 28 And of course, we have the Banach-Tarski paradox.

Theorem n Any two bounded sets X , Y ⊂ R with non-empty interior, for n ≥ 3 are equidecomposable.

Valuations Valuations

3 What Dehn showed is: There are valuations on polytopes in R other than the volume. There are many problems related to Hilbert’s third problem that are still open. The most important one is :

Question Does the Dehn-Sydler theorem hold in spherical or hyperbolic geometry?

Some related results : Two sets are equidecomposable (as opposed to equidissectable) if they can be partitioned into D2 congruent sets.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 6 / 28 Valuations Valuations

3 What Dehn showed is: There are valuations on polytopes in R other than the volume. There are many problems related to Hilbert’s third problem that are still open. The most important one is :

Question Does the Dehn-Sydler theorem hold in spherical or hyperbolic geometry?

Some related results : Two sets are equidecomposable (as opposed to equidissectable) if they can be partitioned into D2 congruent sets.

Theorem n Any two bounded sets X , Y ⊂ R with non-empty interior, for n ≥ 3 are equidecomposable.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 6 / 28 Given A ∈ Kd , the Steiner polynomial is the polynomial given by

d d X k |A + xB2 | = Sk x . k=0

The coeﬃcients are called Quermassintegrals and are equal to the Vk upto a scaling. Hadwiger’s theorem says

Theorem

The collection of all continuous rigid motion invariant valuations on Kd is a d + 1 dimensional vector space, spanned by the coeﬃcients of the Steiner Polynomial.

In an earlier paper that motivated Hadwiger’s work, Blaschke characterized continuous valuations on Kd that are SL(d) and translation invariant.

Theorem (Blaschke)

Any such valuation is a linear combination of 1 and S0. (Note that S0 is the volume).

The study of valuations that take values in other semigroups is a beautiful and well developed area.

Valuations Hadwiger’s theorem

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 7 / 28 Hadwiger’s theorem says

Theorem

The collection of all continuous rigid motion invariant valuations on Kd is a d + 1 dimensional vector space, spanned by the coeﬃcients of the Steiner Polynomial.

In an earlier paper that motivated Hadwiger’s work, Blaschke characterized continuous valuations on Kd that are SL(d) and translation invariant.

Theorem (Blaschke)

Any such valuation is a linear combination of 1 and S0. (Note that S0 is the volume).

The study of valuations that take values in other semigroups is a beautiful and well developed area.

Valuations Hadwiger’s theorem

Given A ∈ Kd , the Steiner polynomial is the polynomial given by

d d X k |A + xB2 | = Sk x . k=0

The coeﬃcients are called Quermassintegrals and are equal to the Vk upto a scaling.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 7 / 28 In an earlier paper that motivated Hadwiger’s work, Blaschke characterized continuous valuations on Kd that are SL(d) and translation invariant.

Theorem (Blaschke)

Any such valuation is a linear combination of 1 and S0. (Note that S0 is the volume).

The study of valuations that take values in other semigroups is a beautiful and well developed area.

Valuations Hadwiger’s theorem

Given A ∈ Kd , the Steiner polynomial is the polynomial given by

d d X k |A + xB2 | = Sk x . k=0

The coeﬃcients are called Quermassintegrals and are equal to the Vk upto a scaling. Hadwiger’s theorem says

Theorem

The collection of all continuous rigid motion invariant valuations on Kd is a d + 1 dimensional vector space, spanned by the coeﬃcients of the Steiner Polynomial.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 7 / 28 Theorem (Blaschke)

Any such valuation is a linear combination of 1 and S0. (Note that S0 is the volume).

The study of valuations that take values in other semigroups is a beautiful and well developed area.

Valuations Hadwiger’s theorem

Given A ∈ Kd , the Steiner polynomial is the polynomial given by

d d X k |A + xB2 | = Sk x . k=0

The coeﬃcients are called Quermassintegrals and are equal to the Vk upto a scaling. Hadwiger’s theorem says

Theorem

The collection of all continuous rigid motion invariant valuations on Kd is a d + 1 dimensional vector space, spanned by the coeﬃcients of the Steiner Polynomial.

In an earlier paper that motivated Hadwiger’s work, Blaschke characterized continuous valuations on Kd that are SL(d) and translation invariant.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 7 / 28 The study of valuations that take values in other semigroups is a beautiful and well developed area.

Valuations Hadwiger’s theorem

Given A ∈ Kd , the Steiner polynomial is the polynomial given by

d d X k |A + xB2 | = Sk x . k=0

The coeﬃcients are called Quermassintegrals and are equal to the Vk upto a scaling. Hadwiger’s theorem says

Theorem

The collection of all continuous rigid motion invariant valuations on Kd is a d + 1 dimensional vector space, spanned by the coeﬃcients of the Steiner Polynomial.

In an earlier paper that motivated Hadwiger’s work, Blaschke characterized continuous valuations on Kd that are SL(d) and translation invariant.

Theorem (Blaschke)

Any such valuation is a linear combination of 1 and S0. (Note that S0 is the volume).

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 7 / 28 Valuations Hadwiger’s theorem

Given A ∈ Kd , the Steiner polynomial is the polynomial given by

d d X k |A + xB2 | = Sk x . k=0

The coeﬃcients are called Quermassintegrals and are equal to the Vk upto a scaling. Hadwiger’s theorem says

Theorem

The collection of all continuous rigid motion invariant valuations on Kd is a d + 1 dimensional vector space, spanned by the coeﬃcients of the Steiner Polynomial.

In an earlier paper that motivated Hadwiger’s work, Blaschke characterized continuous valuations on Kd that are SL(d) and translation invariant.

Theorem (Blaschke)

Any such valuation is a linear combination of 1 and S0. (Note that S0 is the volume).

The study of valuations that take values in other semigroups is a beautiful and well developed area.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 7 / 28 The natural question here is : What are the translation invariant and SL(d) invariant valuations?

This has a beautiful and clean answer. To do this, let us introduce the Ehrhart polynomial of a d lattice polytope. Let Ld be the set of all lattice polytopes in Z . Theorem (Ehrhart’s theorem)

Let P be a lattice polytope : Then there is a polynomial EhrP of degree equal to the aﬃne dimension of P such that d EhrP (n) = |nP ∩ Z |, where |A| is the number of lattice points in A. This polynomial has rational coeﬃcients.

If we write d X k Ehrp(x) = Ek (P)x , k=0

we see that Ed is the volume and E0 = 1.

Valuations Vaulations on Lattice Polytopes

d d Let L be a lattice in R : We will assume that L = Z .

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 8 / 28 This has a beautiful and clean answer. To do this, let us introduce the Ehrhart polynomial of a d lattice polytope. Let Ld be the set of all lattice polytopes in Z . Theorem (Ehrhart’s theorem)

If we write d X k Ehrp(x) = Ek (P)x , k=0

we see that Ed is the volume and E0 = 1.

Valuations Vaulations on Lattice Polytopes

d d Let L be a lattice in R : We will assume that L = Z . The natural question here is : What are the translation invariant and SL(d) invariant valuations?

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 8 / 28 Theorem (Ehrhart’s theorem)

If we write d X k Ehrp(x) = Ek (P)x , k=0

we see that Ed is the volume and E0 = 1.

Valuations Vaulations on Lattice Polytopes

d d Let L be a lattice in R : We will assume that L = Z . The natural question here is : What are the translation invariant and SL(d) invariant valuations?

This has a beautiful and clean answer. To do this, let us introduce the Ehrhart polynomial of a d lattice polytope. Let Ld be the set of all lattice polytopes in Z .

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 8 / 28 If we write d X k Ehrp(x) = Ek (P)x , k=0

we see that Ed is the volume and E0 = 1.

Valuations Vaulations on Lattice Polytopes

d d Let L be a lattice in R : We will assume that L = Z . The natural question here is : What are the translation invariant and SL(d) invariant valuations?

This has a beautiful and clean answer. To do this, let us introduce the Ehrhart polynomial of a d lattice polytope. Let Ld be the set of all lattice polytopes in Z . Theorem (Ehrhart’s theorem)

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 8 / 28 Valuations Vaulations on Lattice Polytopes

d d Let L be a lattice in R : We will assume that L = Z . The natural question here is : What are the translation invariant and SL(d) invariant valuations?

This has a beautiful and clean answer. To do this, let us introduce the Ehrhart polynomial of a d lattice polytope. Let Ld be the set of all lattice polytopes in Z . Theorem (Ehrhart’s theorem)

If we write d X k Ehrp(x) = Ek (P)x , k=0 we see that Ed is the volume and E0 = 1.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 8 / 28 d+1 Let Sd be the d simplex : The convex hull of e1,..., ed+1 in R . Then d + x (d + x)(d + x − 1) ... (x + 1) Ehr (x) = = . Sd d d!

Surely the coeﬃcients of the Ehrhart polynomial of any lattice are non-negative?

Rather surprisingly this is false. The Reeve Tetrahedron is

Rt := conv{(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, r)}.

This was introduced by John Reeve in 1902 to show that there is no analogue of Pick’s theorem in higher dimensions. There are no lattice points in the Reeve tetrahedron apart from the vertices but the volume is r/6 which can be arbitrarily large.

rt3 r Ehr = + t2 + 2 − t + 1. Rt 6 6

Valuations Some Ehrhart polynomials

Let Bd be the d dimensional box. Then

d EhrBd (x) = (x + 1) .

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 9 / 28 Surely the coeﬃcients of the Ehrhart polynomial of any lattice are non-negative?

Rather surprisingly this is false. The Reeve Tetrahedron is

Rt := conv{(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, r)}.

rt3 r Ehr = + t2 + 2 − t + 1. Rt 6 6

Valuations Some Ehrhart polynomials

Let Bd be the d dimensional box. Then

d EhrBd (x) = (x + 1) .

d+1 Let Sd be the d simplex : The convex hull of e1,..., ed+1 in R . Then d + x (d + x)(d + x − 1) ... (x + 1) Ehr (x) = = . Sd d d!

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 9 / 28 Rather surprisingly this is false. The Reeve Tetrahedron is

Rt := conv{(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, r)}.

rt3 r Ehr = + t2 + 2 − t + 1. Rt 6 6

Valuations Some Ehrhart polynomials

Let Bd be the d dimensional box. Then

d EhrBd (x) = (x + 1) .

d+1 Let Sd be the d simplex : The convex hull of e1,..., ed+1 in R . Then d + x (d + x)(d + x − 1) ... (x + 1) Ehr (x) = = . Sd d d!

Surely the coeﬃcients of the Ehrhart polynomial of any lattice are non-negative?

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 9 / 28 This was introduced by John Reeve in 1902 to show that there is no analogue of Pick’s theorem in higher dimensions. There are no lattice points in the Reeve tetrahedron apart from the vertices but the volume is r/6 which can be arbitrarily large.

rt3 r Ehr = + t2 + 2 − t + 1. Rt 6 6

Valuations Some Ehrhart polynomials

Let Bd be the d dimensional box. Then

d EhrBd (x) = (x + 1) .

d+1 Let Sd be the d simplex : The convex hull of e1,..., ed+1 in R . Then d + x (d + x)(d + x − 1) ... (x + 1) Ehr (x) = = . Sd d d!

Surely the coeﬃcients of the Ehrhart polynomial of any lattice are non-negative?

Rather surprisingly this is false. The Reeve Tetrahedron is

Rt := conv{(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, r)}.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 9 / 28 Valuations Some Ehrhart polynomials

Let Bd be the d dimensional box. Then

d EhrBd (x) = (x + 1) .

d+1 Let Sd be the d simplex : The convex hull of e1,..., ed+1 in R . Then d + x (d + x)(d + x − 1) ... (x + 1) Ehr (x) = = . Sd d d!

Surely the coeﬃcients of the Ehrhart polynomial of any lattice are non-negative?

Rather surprisingly this is false. The Reeve Tetrahedron is

Rt := conv{(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, r)}.

rt3 r Ehr = + t2 + 2 − t + 1. Rt 6 6

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 9 / 28 Stanley in 1974 showed that the usual monomial basis is not the best basis to write the Ehrhart polynomial in.

Theorem (Stanley) ∗ ∗ Let P be a lattice polynomial of dimension d. Then, there are natural numbers h0 ,..., hd such that n + d n + d − 1 n Ehr(n) = h∗ + h∗ + ... + h∗ . 0 d 1 d d d

Note that the elements of the h∗ vector are not valuations : The h∗ vector depends upon the dimension of the polytope.

Valuations The Betke-Kneser theorem

Betke and Kneser proved the following beautiful analogue of Hadwiger’s theorem.

Theorem (Betke-Kneser)

Any SL(d) and translation invariant valuation on Ld is a linear combination of the coeﬃcients of the Ehrhart polynomial.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 10 / 28 Note that the elements of the h∗ vector are not valuations : The h∗ vector depends upon the dimension of the polytope.

Valuations The Betke-Kneser theorem

Betke and Kneser proved the following beautiful analogue of Hadwiger’s theorem.

Theorem (Betke-Kneser)

Any SL(d) and translation invariant valuation on Ld is a linear combination of the coeﬃcients of the Ehrhart polynomial.

Stanley in 1974 showed that the usual monomial basis is not the best basis to write the Ehrhart polynomial in.

Theorem (Stanley) ∗ ∗ Let P be a lattice polynomial of dimension d. Then, there are natural numbers h0 ,..., hd such that n + d n + d − 1 n Ehr(n) = h∗ + h∗ + ... + h∗ . 0 d 1 d d d

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 10 / 28 Valuations The Betke-Kneser theorem

Betke and Kneser proved the following beautiful analogue of Hadwiger’s theorem.

Theorem (Betke-Kneser)

Any SL(d) and translation invariant valuation on Ld is a linear combination of the coeﬃcients of the Ehrhart polynomial.

Stanley in 1974 showed that the usual monomial basis is not the best basis to write the Ehrhart polynomial in.

Theorem (Stanley) ∗ ∗ Let P be a lattice polynomial of dimension d. Then, there are natural numbers h0 ,..., hd such that n + d n + d − 1 n Ehr(n) = h∗ + h∗ + ... + h∗ . 0 d 1 d d d

Note that the elements of the h∗ vector are not valuations : The h∗ vector depends upon the dimension of the polytope.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 10 / 28 Theorem (McMullen)

Let φ : Ld → R be a translation invariant valuation. Then there is a polynomial ϕP of degree at most the aﬃne dimension of P such that

ϕP (n) = φ(nP), ∀n ∈ N ∪ {0}.

The theorem also holds when we replace the range space R by any group G. (The notion of a polynomial is appropriately deﬁned). d It also works when we replace lattice polytopes by general polytopes and consider R translation invariant valuations.

Valuations McMullen’s theorem

In 1977, Peter McMullen proved a far reaching generalization of the Betke-Kneser theorem.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 11 / 28 The theorem also holds when we replace the range space R by any group G. (The notion of a polynomial is appropriately deﬁned). d It also works when we replace lattice polytopes by general polytopes and consider R translation invariant valuations.

Valuations McMullen’s theorem

In 1977, Peter McMullen proved a far reaching generalization of the Betke-Kneser theorem.

Theorem (McMullen)

Let φ : Ld → R be a translation invariant valuation. Then there is a polynomial ϕP of degree at most the aﬃne dimension of P such that

ϕP (n) = φ(nP), ∀n ∈ N ∪ {0}.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 11 / 28 Valuations McMullen’s theorem

In 1977, Peter McMullen proved a far reaching generalization of the Betke-Kneser theorem.

Theorem (McMullen)

Let φ : Ld → R be a translation invariant valuation. Then there is a polynomial ϕP of degree at most the aﬃne dimension of P such that

ϕP (n) = φ(nP), ∀n ∈ N ∪ {0}.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 11 / 28 Valuations McMullen’s theorem

In 1977, Peter McMullen proved a far reaching generalization of the Betke-Kneser theorem.

Theorem (McMullen)

Let φ : Ld → R be a translation invariant valuation. Then there is a polynomial ϕP of degree at most the aﬃne dimension of P such that

ϕP (n) = φ(nP), ∀n ∈ N ∪ {0}.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 11 / 28 Instead of considering valuations on convex sets, may we consider valuations on the larger class of PolyConvex sets.

Definition A relatively open polyhedron is a polyhedral convex set that is open in the affine space that it lies in. A polyconvex set is a finite union of relatively open polyhedra.

In practical terms, a polyconvex set is a disjoint union of ﬁnitely many sets that are polyhedra with the interiors of certain facets removed.

The most important valuation by far is the following.

Theorem d There is a unique integer valued valuation on polyconvex sets in R which assigns the value 1 to every (closed) polytope. This valuation is called the Euler Characteristic.

Valuations The Euler Characteristic

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 12 / 28 In practical terms, a polyconvex set is a disjoint union of ﬁnitely many sets that are polyhedra with the interiors of certain facets removed.

The most important valuation by far is the following.

Theorem d There is a unique integer valued valuation on polyconvex sets in R which assigns the value 1 to every (closed) polytope. This valuation is called the Euler Characteristic.

Valuations The Euler Characteristic

Instead of considering valuations on convex sets, may we consider valuations on the larger class of PolyConvex sets.

Definition A relatively open polyhedron is a polyhedral convex set that is open in the affine space that it lies in. A polyconvex set is a finite union of relatively open polyhedra.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 12 / 28 The most important valuation by far is the following.

Theorem d There is a unique integer valued valuation on polyconvex sets in R which assigns the value 1 to every (closed) polytope. This valuation is called the Euler Characteristic.

Valuations The Euler Characteristic

Instead of considering valuations on convex sets, may we consider valuations on the larger class of PolyConvex sets.

Definition A relatively open polyhedron is a polyhedral convex set that is open in the affine space that it lies in. A polyconvex set is a finite union of relatively open polyhedra.

In practical terms, a polyconvex set is a disjoint union of ﬁnitely many sets that are polyhedra with the interiors of certain facets removed.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 12 / 28 Valuations The Euler Characteristic

Instead of considering valuations on convex sets, may we consider valuations on the larger class of PolyConvex sets.

Definition A relatively open polyhedron is a polyhedral convex set that is open in the affine space that it lies in. A polyconvex set is a finite union of relatively open polyhedra.

In practical terms, a polyconvex set is a disjoint union of ﬁnitely many sets that are polyhedra with the interiors of certain facets removed.

The most important valuation by far is the following.

Theorem d There is a unique integer valued valuation on polyconvex sets in R which assigns the value 1 to every (closed) polytope. This valuation is called the Euler Characteristic.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 12 / 28 Valuations The Euler Characteristic

Instead of considering valuations on convex sets, may we consider valuations on the larger class of PolyConvex sets.

Definition A relatively open polyhedron is a polyhedral convex set that is open in the affine space that it lies in. A polyconvex set is a finite union of relatively open polyhedra.

In practical terms, a polyconvex set is a disjoint union of ﬁnitely many sets that are polyhedra with the interiors of certain facets removed.

The most important valuation by far is the following.

Theorem d There is a unique integer valued valuation on polyconvex sets in R which assigns the value 1 to every (closed) polytope. This valuation is called the Euler Characteristic.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 12 / 28 Valuations The Euler Characteristic

Instead of considering valuations on convex sets, may we consider valuations on the larger class of PolyConvex sets.

Definition A relatively open polyhedron is a polyhedral convex set that is open in the affine space that it lies in. A polyconvex set is a finite union of relatively open polyhedra.

In practical terms, a polyconvex set is a disjoint union of ﬁnitely many sets that are polyhedra with the interiors of certain facets removed.

The most important valuation by far is the following.

Theorem d There is a unique integer valued valuation on polyconvex sets in R which assigns the value 1 to every (closed) polytope. This valuation is called the Euler Characteristic.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 12 / 28 Theorem (Macdonald)

Let P ∈ Ld . Then dim(P) EhrP (−n) = (−1) EhrPo (n), where Po is the relative interior of P. This was conjectured by Ehrhart and proved by him in several cases. The full proof was given by Ian Macdonald in 1971.

This theorem was generalized by McMullen to all translation invariant valuations.

Theorem (McMullen)

Let φ be a valuation on Ld . Then

X dim(F ) ϕP (−n) = (−1) ϕF (n). F ⊂P

Valuations Combinatorial Reciprocity

The Ehrhart polynomial of a lattice polytope, EhrP makes geometric sense when evaluated on the non-negative integers. It is a fundamental fact that Ehrp(−n) has a natural meaning as well.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 13 / 28 This theorem was generalized by McMullen to all translation invariant valuations.

Theorem (McMullen)

Let φ be a valuation on Ld . Then

X dim(F ) ϕP (−n) = (−1) ϕF (n). F ⊂P

Valuations Combinatorial Reciprocity

The Ehrhart polynomial of a lattice polytope, EhrP makes geometric sense when evaluated on the non-negative integers. It is a fundamental fact that Ehrp(−n) has a natural meaning as well.

Theorem (Macdonald)

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 13 / 28 Valuations Combinatorial Reciprocity

The Ehrhart polynomial of a lattice polytope, EhrP makes geometric sense when evaluated on the non-negative integers. It is a fundamental fact that Ehrp(−n) has a natural meaning as well.

Theorem (Macdonald)

This theorem was generalized by McMullen to all translation invariant valuations.

Theorem (McMullen)

Let φ be a valuation on Ld . Then

X dim(F ) ϕP (−n) = (−1) ϕF (n). F ⊂P

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 13 / 28 This is a well studied problem and an active current area of research. We summarize some results here.

To any polytope, we may consider its lattice of faces, that is the inclusion ordered Poset of faces. Two polytopes are said to be combinatorially equivalent if they have isomorphic face lattices.

Theorem (Liu, 2009) Any lattice polytope is combinatorially equivalent to a lattice polytope that is Ehrhart positive.

For instance, the Reeve tetrahedron and the standard tetrahedron are combinatorially equivalent and the latter is easily seen to be Ehrhart positive (by a simple explicit computation for instance).

This shows that Ehrhart Positivity is not a combinatorial property, but a a geometric one.

Ehrhart Positivity Ehrhart Positivity

Recall the deﬁnition of the Ehrhart polynomial,

d EhrP (n) = |nP ∩ Z |,

Question Which lattice polytopes have Ehrhart polynomials with non-negative coeﬃcients?

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 14 / 28 To any polytope, we may consider its lattice of faces, that is the inclusion ordered Poset of faces. Two polytopes are said to be combinatorially equivalent if they have isomorphic face lattices.

Theorem (Liu, 2009) Any lattice polytope is combinatorially equivalent to a lattice polytope that is Ehrhart positive.

For instance, the Reeve tetrahedron and the standard tetrahedron are combinatorially equivalent and the latter is easily seen to be Ehrhart positive (by a simple explicit computation for instance).

This shows that Ehrhart Positivity is not a combinatorial property, but a a geometric one.

Ehrhart Positivity Ehrhart Positivity

Recall the deﬁnition of the Ehrhart polynomial,

d EhrP (n) = |nP ∩ Z |,

Question Which lattice polytopes have Ehrhart polynomials with non-negative coeﬃcients?

This is a well studied problem and an active current area of research. We summarize some results here.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 14 / 28 For instance, the Reeve tetrahedron and the standard tetrahedron are combinatorially equivalent and the latter is easily seen to be Ehrhart positive (by a simple explicit computation for instance).

This shows that Ehrhart Positivity is not a combinatorial property, but a a geometric one.

Ehrhart Positivity Ehrhart Positivity

Recall the deﬁnition of the Ehrhart polynomial,

d EhrP (n) = |nP ∩ Z |,

Question Which lattice polytopes have Ehrhart polynomials with non-negative coeﬃcients?

This is a well studied problem and an active current area of research. We summarize some results here.

To any polytope, we may consider its lattice of faces, that is the inclusion ordered Poset of faces. Two polytopes are said to be combinatorially equivalent if they have isomorphic face lattices.

Theorem (Liu, 2009) Any lattice polytope is combinatorially equivalent to a lattice polytope that is Ehrhart positive.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 14 / 28 This shows that Ehrhart Positivity is not a combinatorial property, but a a geometric one.

Ehrhart Positivity Ehrhart Positivity

Recall the deﬁnition of the Ehrhart polynomial,

d EhrP (n) = |nP ∩ Z |,

Question Which lattice polytopes have Ehrhart polynomials with non-negative coeﬃcients?

This is a well studied problem and an active current area of research. We summarize some results here.

To any polytope, we may consider its lattice of faces, that is the inclusion ordered Poset of faces. Two polytopes are said to be combinatorially equivalent if they have isomorphic face lattices.

Theorem (Liu, 2009) Any lattice polytope is combinatorially equivalent to a lattice polytope that is Ehrhart positive.

For instance, the Reeve tetrahedron and the standard tetrahedron are combinatorially equivalent and the latter is easily seen to be Ehrhart positive (by a simple explicit computation for instance).

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 14 / 28 Ehrhart Positivity Ehrhart Positivity

Recall the deﬁnition of the Ehrhart polynomial,

d EhrP (n) = |nP ∩ Z |,

Question Which lattice polytopes have Ehrhart polynomials with non-negative coeﬃcients?

This is a well studied problem and an active current area of research. We summarize some results here.

To any polytope, we may consider its lattice of faces, that is the inclusion ordered Poset of faces. Two polytopes are said to be combinatorially equivalent if they have isomorphic face lattices.

Theorem (Liu, 2009) Any lattice polytope is combinatorially equivalent to a lattice polytope that is Ehrhart positive.

For instance, the Reeve tetrahedron and the standard tetrahedron are combinatorially equivalent and the latter is easily seen to be Ehrhart positive (by a simple explicit computation for instance).

This shows that Ehrhart Positivity is not a combinatorial property, but a a geometric one.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 14 / 28 The larger class of Flow Polytopes are also Ehrhart positive. This fact uses a formula for the Ehrhart polynomial in terms of the Kostant partition functions from representation theory.

Crosspolytopes and several derived polytopes are Ehrhart positive : This uses an interesting fact, namely that their Ehrhart polynomials have roots on S1. That this implies positivity is easy to see.

Ehrhart Positivity Classes of Ehrhart Positive Polytopes

d Given a vector a = (a1,..., ad ) ∈ N , the associated Stanley-Pitman polytope is

i i d X X PSd (a) = {x ∈ R≥0 | xi ≤ ai , i ∈ [d]}. j=1 j=1

Pitman and Stanley gave an exact formula for the Ehrhart polynomial, writing it as a sum of a product of binomial coeﬃcients.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 15 / 28 Crosspolytopes and several derived polytopes are Ehrhart positive : This uses an interesting fact, namely that their Ehrhart polynomials have roots on S1. That this implies positivity is easy to see.

Ehrhart Positivity Classes of Ehrhart Positive Polytopes

d Given a vector a = (a1,..., ad ) ∈ N , the associated Stanley-Pitman polytope is

i i d X X PSd (a) = {x ∈ R≥0 | xi ≤ ai , i ∈ [d]}. j=1 j=1

Pitman and Stanley gave an exact formula for the Ehrhart polynomial, writing it as a sum of a product of binomial coeﬃcients.

The larger class of Flow Polytopes are also Ehrhart positive. This fact uses a formula for the Ehrhart polynomial in terms of the Kostant partition functions from representation theory.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 15 / 28 Ehrhart Positivity Classes of Ehrhart Positive Polytopes

d Given a vector a = (a1,..., ad ) ∈ N , the associated Stanley-Pitman polytope is

i i d X X PSd (a) = {x ∈ R≥0 | xi ≤ ai , i ∈ [d]}. j=1 j=1

Pitman and Stanley gave an exact formula for the Ehrhart polynomial, writing it as a sum of a product of binomial coeﬃcients.

The larger class of Flow Polytopes are also Ehrhart positive. This fact uses a formula for the Ehrhart polynomial in terms of the Kostant partition functions from representation theory.

Crosspolytopes and several derived polytopes are Ehrhart positive : This uses an interesting fact, namely that their Ehrhart polynomials have roots on S1. That this implies positivity is easy to see.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 15 / 28 Theorem (Stanley) k The coeﬃcient of t in EhrZ is X h(X ), X where X is the collection of all linearly independent size k subsets and h(X ) is the gcd of all k × k minors of the matrix whose column vectors are the elements in X . The most important Zonotope (and perhaps polytope) is the Regular Permutohedron, X Πd = conv{(σ(1), . . . , σ(d + 1)) : σ ∈ Sd+1} = [0, ej − ei ]. i

Stanley’s theorem specializes to :EZ (k) is the # forests on [d + 1] with exactly d + 1 − k trees.

Ehrhart Positivity Zonotopes

Zonotopes are Minkowski sums of line segments,

m X Z = [0, vi ], i=1

d where the vi ∈ Z . In this case, the coeﬃcients of the Ehrhart polynomial have meaning.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 16 / 28 The most important Zonotope (and perhaps polytope) is the Regular Permutohedron, X Πd = conv{(σ(1), . . . , σ(d + 1)) : σ ∈ Sd+1} = [0, ej − ei ]. i

Stanley’s theorem specializes to :EZ (k) is the # forests on [d + 1] with exactly d + 1 − k trees.

Ehrhart Positivity Zonotopes

Zonotopes are Minkowski sums of line segments,

m X Z = [0, vi ], i=1

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 16 / 28 Stanley’s theorem specializes to :EZ (k) is the # forests on [d + 1] with exactly d + 1 − k trees.

Ehrhart Positivity Zonotopes

Zonotopes are Minkowski sums of line segments,

m X Z = [0, vi ], i=1

d where the vi ∈ Z . In this case, the coeﬃcients of the Ehrhart polynomial have meaning. Theorem (Stanley) k The coeﬃcient of t in EhrZ is X h(X ), X where X is the collection of all linearly independent size k subsets and h(X ) is the gcd of all k × k minors of the matrix whose column vectors are the elements in X . The most important Zonotope (and perhaps polytope) is the Regular Permutohedron, X Πd = conv{(σ(1), . . . , σ(d + 1)) : σ ∈ Sd+1} = [0, ej − ei ]. i

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 16 / 28 Ehrhart Positivity Zonotopes

Zonotopes are Minkowski sums of line segments,

m X Z = [0, vi ], i=1

Stanley’s theorem specializes to :EZ (k) is the # forests on [d + 1] with exactly d + 1 − k trees.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 16 / 28 All maximal independent sets in a matroid have the same cardinality and these sets are called the bases of the matroid and denoted by B(M).

The matroid (base) polytope is PM = conv{e ∈ B}.

For instance, the following are Linear Programs over Matroid Polytopes. 1 The Travelling Salesman problem, 2 Finding maximum weight matchings in graphs, 3 Finding maximum ﬂows in graphs,

Theorem (Fundamental Meta-theorem of Combinatorial Optimization) Optimization problems where the associated matroid polytopes have compact facet description are tractable. If a problem is intractable, then the corresponding matroid polytope has complex facet structure.

Generalized Permutohedra Matroid Polytopes

A matroid M is a ﬁnite set X and a collection of subsets T (called independent sets) which are 1 Downward closed. 2 For every e ∈ T and i ∈ X \ e, there is a j ∈ e such that e ∪ i \{j} ∈ T .

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 17 / 28 The matroid (base) polytope is PM = conv{e ∈ B}.

For instance, the following are Linear Programs over Matroid Polytopes. 1 The Travelling Salesman problem, 2 Finding maximum weight matchings in graphs, 3 Finding maximum ﬂows in graphs,

Generalized Permutohedra Matroid Polytopes

A matroid M is a ﬁnite set X and a collection of subsets T (called independent sets) which are 1 Downward closed. 2 For every e ∈ T and i ∈ X \ e, there is a j ∈ e such that e ∪ i \{j} ∈ T . All maximal independent sets in a matroid have the same cardinality and these sets are called the bases of the matroid and denoted by B(M).

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 17 / 28 For instance, the following are Linear Programs over Matroid Polytopes. 1 The Travelling Salesman problem, 2 Finding maximum weight matchings in graphs, 3 Finding maximum ﬂows in graphs,

Generalized Permutohedra Matroid Polytopes

The matroid (base) polytope is PM = conv{e ∈ B}.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 17 / 28 Theorem (Fundamental Meta-theorem of Combinatorial Optimization) Optimization problems where the associated matroid polytopes have compact facet description are tractable. If a problem is intractable, then the corresponding matroid polytope has complex facet structure.

Generalized Permutohedra Matroid Polytopes

The matroid (base) polytope is PM = conv{e ∈ B}.

For instance, the following are Linear Programs over Matroid Polytopes. 1 The Travelling Salesman problem, 2 Finding maximum weight matchings in graphs, 3 Finding maximum ﬂows in graphs,

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 17 / 28 Generalized Permutohedra Matroid Polytopes

The matroid (base) polytope is PM = conv{e ∈ B}.

For instance, the following are Linear Programs over Matroid Polytopes. 1 The Travelling Salesman problem, 2 Finding maximum weight matchings in graphs, 3 Finding maximum ﬂows in graphs,

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 17 / 28 In 2007, Alexander Postnikov came up with a highly inﬂuential class of polytopes, which he called Generalized Permutohedra. Deﬁnition (Regular and Generalized Permutohedra) d+1 Given α = (α1, . . . , αd+1) ∈ R , the associated permutohedron is

Πα = conv{(ασ(1), . . . , ασ(d+1)) : σ ∈ Sd+1}.

A generalized permutohedron is a polytope obtained by making parallel translates of facets of a regular permutohedron. Note that some vertices may vanish.

An alternate deﬁnition is the following : A generalized permutohedron is any polytope whose normal fan is the coarsening of the normal fan of a permutohedron.

Yet another deﬁnition is the following : A polytope P is a generalized permutohedron if there is another polytopes Q such that P + Q = λΠd . In other words, P is a weak Minkowski summand of Πd .

Generalized Permutohedra Generalized Permutohedra

Conjecture (De Leora et. al. 2007) Matroid polytopes are Ehrhart Positive.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 18 / 28 Deﬁnition (Regular and Generalized Permutohedra) d+1 Given α = (α1, . . . , αd+1) ∈ R , the associated permutohedron is

Πα = conv{(ασ(1), . . . , ασ(d+1)) : σ ∈ Sd+1}.

A generalized permutohedron is a polytope obtained by making parallel translates of facets of a regular permutohedron. Note that some vertices may vanish.

An alternate deﬁnition is the following : A generalized permutohedron is any polytope whose normal fan is the coarsening of the normal fan of a permutohedron.

Yet another deﬁnition is the following : A polytope P is a generalized permutohedron if there is another polytopes Q such that P + Q = λΠd . In other words, P is a weak Minkowski summand of Πd .

Generalized Permutohedra Generalized Permutohedra

Conjecture (De Leora et. al. 2007) Matroid polytopes are Ehrhart Positive.

In 2007, Alexander Postnikov came up with a highly inﬂuential class of polytopes, which he called Generalized Permutohedra.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 18 / 28 An alternate deﬁnition is the following : A generalized permutohedron is any polytope whose normal fan is the coarsening of the normal fan of a permutohedron.

Yet another deﬁnition is the following : A polytope P is a generalized permutohedron if there is another polytopes Q such that P + Q = λΠd . In other words, P is a weak Minkowski summand of Πd .

Generalized Permutohedra Generalized Permutohedra

Conjecture (De Leora et. al. 2007) Matroid polytopes are Ehrhart Positive.

In 2007, Alexander Postnikov came up with a highly inﬂuential class of polytopes, which he called Generalized Permutohedra. Deﬁnition (Regular and Generalized Permutohedra) d+1 Given α = (α1, . . . , αd+1) ∈ R , the associated permutohedron is

Πα = conv{(ασ(1), . . . , ασ(d+1)) : σ ∈ Sd+1}.

A generalized permutohedron is a polytope obtained by making parallel translates of facets of a regular permutohedron. Note that some vertices may vanish.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 18 / 28 Yet another deﬁnition is the following : A polytope P is a generalized permutohedron if there is another polytopes Q such that P + Q = λΠd . In other words, P is a weak Minkowski summand of Πd .

Generalized Permutohedra Generalized Permutohedra

Conjecture (De Leora et. al. 2007) Matroid polytopes are Ehrhart Positive.

Πα = conv{(ασ(1), . . . , ασ(d+1)) : σ ∈ Sd+1}.

A generalized permutohedron is a polytope obtained by making parallel translates of facets of a regular permutohedron. Note that some vertices may vanish.

An alternate deﬁnition is the following : A generalized permutohedron is any polytope whose normal fan is the coarsening of the normal fan of a permutohedron.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 18 / 28 Generalized Permutohedra Generalized Permutohedra

Conjecture (De Leora et. al. 2007) Matroid polytopes are Ehrhart Positive.

Πα = conv{(ασ(1), . . . , ασ(d+1)) : σ ∈ Sd+1}.

A generalized permutohedron is a polytope obtained by making parallel translates of facets of a regular permutohedron. Note that some vertices may vanish.

An alternate deﬁnition is the following : A generalized permutohedron is any polytope whose normal fan is the coarsening of the normal fan of a permutohedron.

Yet another deﬁnition is the following : A polytope P is a generalized permutohedron if there is another polytopes Q such that P + Q = λΠd . In other words, P is a weak Minkowski summand of Πd .

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 18 / 28 Deﬁnition A function f deﬁned on subsets of [d] is called submodular if for any S ⊂ T ⊂ [n] and i not in T ,

f (T ∪ {i}) − f (T ) ≤ f (S ∪ {i}) − f (S).

A function is supermodular if the reverse inequality holds.

2[d] For every vector {zI }I ⊆[d] ∈ R with z∅ = 0 let

 d   d X X  P({zI }) = x ∈ R : xi = z[d] , xi ≥ zI for all ∅ ⊆ I ⊂ [d] ,  i=1 i∈I 

Theorem (Postnikov)

The polytope P({zI }) is a generalized permutahedron if and only if the {zI }I ∈[d] deﬁne a [d] supermodular function 2 → R.

Theorem (Ardila, Benedetti, Doker) Matroid base polytopes are Generalized Permutohedra.

Generalized Permutohedra Submodular functions

Another very useful characterization of generalized permutohedra uses submodular functions.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 19 / 28 2[d] For every vector {zI }I ⊆[d] ∈ R with z∅ = 0 let

 d   d X X  P({zI }) = x ∈ R : xi = z[d] , xi ≥ zI for all ∅ ⊆ I ⊂ [d] ,  i=1 i∈I 

Theorem (Postnikov)

The polytope P({zI }) is a generalized permutahedron if and only if the {zI }I ∈[d] deﬁne a [d] supermodular function 2 → R.

Theorem (Ardila, Benedetti, Doker) Matroid base polytopes are Generalized Permutohedra.

Generalized Permutohedra Submodular functions

Another very useful characterization of generalized permutohedra uses submodular functions.

Deﬁnition A function f deﬁned on subsets of [d] is called submodular if for any S ⊂ T ⊂ [n] and i not in T ,

f (T ∪ {i}) − f (T ) ≤ f (S ∪ {i}) − f (S).

A function is supermodular if the reverse inequality holds.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 19 / 28 Theorem (Postnikov)

The polytope P({zI }) is a generalized permutahedron if and only if the {zI }I ∈[d] deﬁne a [d] supermodular function 2 → R.

Theorem (Ardila, Benedetti, Doker) Matroid base polytopes are Generalized Permutohedra.

Generalized Permutohedra Submodular functions

Another very useful characterization of generalized permutohedra uses submodular functions.

Deﬁnition A function f deﬁned on subsets of [d] is called submodular if for any S ⊂ T ⊂ [n] and i not in T ,

f (T ∪ {i}) − f (T ) ≤ f (S ∪ {i}) − f (S).

A function is supermodular if the reverse inequality holds.

2[d] For every vector {zI }I ⊆[d] ∈ R with z∅ = 0 let

 d   d X X  P({zI }) = x ∈ R : xi = z[d] , xi ≥ zI for all ∅ ⊆ I ⊂ [d] ,  i=1 i∈I 

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 19 / 28 Theorem (Ardila, Benedetti, Doker) Matroid base polytopes are Generalized Permutohedra.

Generalized Permutohedra Submodular functions

Another very useful characterization of generalized permutohedra uses submodular functions.

Deﬁnition A function f deﬁned on subsets of [d] is called submodular if for any S ⊂ T ⊂ [n] and i not in T ,

f (T ∪ {i}) − f (T ) ≤ f (S ∪ {i}) − f (S).

A function is supermodular if the reverse inequality holds.

2[d] For every vector {zI }I ⊆[d] ∈ R with z∅ = 0 let

 d   d X X  P({zI }) = x ∈ R : xi = z[d] , xi ≥ zI for all ∅ ⊆ I ⊂ [d] ,  i=1 i∈I 

Theorem (Postnikov)

The polytope P({zI }) is a generalized permutahedron if and only if the {zI }I ∈[d] deﬁne a [d] supermodular function 2 → R.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 19 / 28 Generalized Permutohedra Submodular functions

Another very useful characterization of generalized permutohedra uses submodular functions.

Deﬁnition A function f deﬁned on subsets of [d] is called submodular if for any S ⊂ T ⊂ [n] and i not in T ,

f (T ∪ {i}) − f (T ) ≤ f (S ∪ {i}) − f (S).

A function is supermodular if the reverse inequality holds.

2[d] For every vector {zI }I ⊆[d] ∈ R with z∅ = 0 let

 d   d X X  P({zI }) = x ∈ R : xi = z[d] , xi ≥ zI for all ∅ ⊆ I ⊂ [d] ,  i=1 i∈I 

Theorem (Postnikov)

The polytope P({zI }) is a generalized permutahedron if and only if the {zI }I ∈[d] deﬁne a [d] supermodular function 2 → R.

Theorem (Ardila, Benedetti, Doker) Matroid base polytopes are Generalized Permutohedra.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 19 / 28 Let ∆∅ = {0} and for ∅ 6= I ⊆ [d] let

∆I = {ei : i ∈ I }

d be the standard simplices where e1,..., ed are the standard basis vectors in R . Theorem (Jochemko, Ravichandran, 2019)

Let {yI }I ∈[d] be a vector of real numbers. Then the following are equivalent. P (i) The signed Minkowski sum I ⊆[d] yI ∆I deﬁnes a generalized permutahedron. [d] (ii) For all 2-element subsets E ∈ 2 and all T ⊆ [d] such that E ⊆ T X yI ≥ 0 . (3.1) E⊆I ⊆T

Further, every generalized permutohedron is of the above form. In particular, the collection P of all coefficients {yI }I ∈[d] such that I ⊆[d] yI ∆I defines a generalized permutahedron is a polyhedral cone. The inequalities are facet-defining.

This theorem uses the supermodular characterization of Permutohedra due to Postnikov together with a theorem of Schneider on facets of Minkowski sums of polytopes.

Generalized Permutohedra Characterization of Generalized Permutohedra via signed Minkowski sums

Our proof uses three ideas. The ﬁrst is a representation theorem for Generalized Permutohedra.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 20 / 28 Theorem (Jochemko, Ravichandran, 2019)

This theorem uses the supermodular characterization of Permutohedra due to Postnikov together with a theorem of Schneider on facets of Minkowski sums of polytopes.

Generalized Permutohedra Characterization of Generalized Permutohedra via signed Minkowski sums

Our proof uses three ideas. The ﬁrst is a representation theorem for Generalized Permutohedra.

Let ∆∅ = {0} and for ∅ 6= I ⊆ [d] let

∆I = {ei : i ∈ I }

d be the standard simplices where e1,..., ed are the standard basis vectors in R .

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 20 / 28 This theorem uses the supermodular characterization of Permutohedra due to Postnikov together with a theorem of Schneider on facets of Minkowski sums of polytopes.

Generalized Permutohedra Characterization of Generalized Permutohedra via signed Minkowski sums

Our proof uses three ideas. The ﬁrst is a representation theorem for Generalized Permutohedra.

Let ∆∅ = {0} and for ∅ 6= I ⊆ [d] let

∆I = {ei : i ∈ I }

d be the standard simplices where e1,..., ed are the standard basis vectors in R . Theorem (Jochemko, Ravichandran, 2019)

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 20 / 28 Generalized Permutohedra Characterization of Generalized Permutohedra via signed Minkowski sums

Our proof uses three ideas. The ﬁrst is a representation theorem for Generalized Permutohedra.

Let ∆∅ = {0} and for ∅ 6= I ⊆ [d] let

∆I = {ei : i ∈ I }

d be the standard simplices where e1,..., ed are the standard basis vectors in R . Theorem (Jochemko, Ravichandran, 2019)

This theorem uses the supermodular characterization of Permutohedra due to Postnikov together with a theorem of Schneider on facets of Minkowski sums of polytopes.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 20 / 28 Theorem (Ludwig) The linear term of the Ehrhart polynomial is Minkowski linear.

The proof of this uses a deep theorem of McMullen that is a generalization of Ehrhart’s theorem

Theorem (McMullen) d Given lattice polytopes P1,..., Pm ∈ Z , the function m (Z≥0) 3 (k1,..., km) → |k1P1 + ... + kmPm| agrees with a polynomial.

With this in hand, we can calculate the linear term of the Ehrhart polynomial of any generalized Permutohedron.

The linear term of the n simplex is easily calculated to be 1 1 E(∆i+1) = 1 + + ··· + =: hi 2 i

Generalized Permutohedra The Linear term is Minkowski Linear

Our second contribution exploits a fundamental property of the linear term

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 21 / 28 The proof of this uses a deep theorem of McMullen that is a generalization of Ehrhart’s theorem

Theorem (McMullen) d Given lattice polytopes P1,..., Pm ∈ Z , the function m (Z≥0) 3 (k1,..., km) → |k1P1 + ... + kmPm| agrees with a polynomial.

With this in hand, we can calculate the linear term of the Ehrhart polynomial of any generalized Permutohedron.

The linear term of the n simplex is easily calculated to be 1 1 E(∆i+1) = 1 + + ··· + =: hi 2 i

Generalized Permutohedra The Linear term is Minkowski Linear

Our second contribution exploits a fundamental property of the linear term

Theorem (Ludwig) The linear term of the Ehrhart polynomial is Minkowski linear.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 21 / 28 With this in hand, we can calculate the linear term of the Ehrhart polynomial of any generalized Permutohedron.

The linear term of the n simplex is easily calculated to be 1 1 E(∆i+1) = 1 + + ··· + =: hi 2 i

Generalized Permutohedra The Linear term is Minkowski Linear

Our second contribution exploits a fundamental property of the linear term

Theorem (Ludwig) The linear term of the Ehrhart polynomial is Minkowski linear.

The proof of this uses a deep theorem of McMullen that is a generalization of Ehrhart’s theorem

Theorem (McMullen) d Given lattice polytopes P1,..., Pm ∈ Z , the function m (Z≥0) 3 (k1,..., km) → |k1P1 + ... + kmPm| agrees with a polynomial.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 21 / 28 The linear term of the n simplex is easily calculated to be 1 1 E(∆i+1) = 1 + + ··· + =: hi 2 i

Generalized Permutohedra The Linear term is Minkowski Linear

Our second contribution exploits a fundamental property of the linear term

Theorem (Ludwig) The linear term of the Ehrhart polynomial is Minkowski linear.

The proof of this uses a deep theorem of McMullen that is a generalization of Ehrhart’s theorem

Theorem (McMullen) d Given lattice polytopes P1,..., Pm ∈ Z , the function m (Z≥0) 3 (k1,..., km) → |k1P1 + ... + kmPm| agrees with a polynomial.

With this in hand, we can calculate the linear term of the Ehrhart polynomial of any generalized Permutohedron.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 21 / 28 Generalized Permutohedra The Linear term is Minkowski Linear

Our second contribution exploits a fundamental property of the linear term

Theorem (Ludwig) The linear term of the Ehrhart polynomial is Minkowski linear.

The proof of this uses a deep theorem of McMullen that is a generalization of Ehrhart’s theorem

Theorem (McMullen) d Given lattice polytopes P1,..., Pm ∈ Z , the function m (Z≥0) 3 (k1,..., km) → |k1P1 + ... + kmPm| agrees with a polynomial.

With this in hand, we can calculate the linear term of the Ehrhart polynomial of any generalized Permutohedron.

The linear term of the n simplex is easily calculated to be 1 1 E(∆i+1) = 1 + + ··· + =: hi 2 i

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 21 / 28 The proof is not diﬃcult : It essentially uses Conic Duality.

Generalized Permutohedra Translation Invariant valuations on Generalized Permutohedra

[d] T For any 2-element subset E ∈ 2 and any T ⊆ [d] such that E ⊆ T let vE be the Minkowski linear functional deﬁned by ( 1 if E ⊆ I ⊆ T , v T (∆ ) = E I 0 otherwise.

We characterize all positive, translation-invariant Minkowski linear functionals on Pd . Proposition

Let ϕ: Pd → R be a Minkowski linear functional. Then ϕ is positive and translation-invariant if T and only if there are nonnegative real numbers cE such that

X X T T ϕ = cE vE . [d] T ⊇E E∈ 2

In particular, the family of positive, translation-invariant Minkowski linear functionals is a T polyhedral cone with rays vE .

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 22 / 28 Generalized Permutohedra Translation Invariant valuations on Generalized Permutohedra

[d] T For any 2-element subset E ∈ 2 and any T ⊆ [d] such that E ⊆ T let vE be the Minkowski linear functional deﬁned by ( 1 if E ⊆ I ⊆ T , v T (∆ ) = E I 0 otherwise.

We characterize all positive, translation-invariant Minkowski linear functionals on Pd . Proposition

Let ϕ: Pd → R be a Minkowski linear functional. Then ϕ is positive and translation-invariant if T and only if there are nonnegative real numbers cE such that

X X T T ϕ = cE vE . [d] T ⊇E E∈ 2

In particular, the family of positive, translation-invariant Minkowski linear functionals is a T polyhedral cone with rays vE . The proof is not diﬃcult : It essentially uses Conic Duality.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 22 / 28 For all 1 ≤ k ≤ d − 1 let fk : Pd → R be the symmetric, translation-invariant Minkowski linear functional deﬁned by i + 1d − i − 1 (f )(∆i+1) = (3.2) k 2 k − i for all 1 ≤ i ≤ d − 1.

Theorem Let ϕ: Pd → R be a Minkowski linear functional. Then ϕ is positive, translation-invariant and symmetric if and only if there are real numbers c1,..., cd−1 ≥ 0 such that

d−1 X ϕ = ck fk . k=1

In particular, the family of all positive, Minkowski linear, translation- and symmetric functionals form a simplicial cone of dimension d − 1.

Generalized Permutohedra Symmetric Minkowski Linear Functionals

The Final step is to specialize the above theorem to the case when the functionals are also symmetric (under co-ordinate permutations).

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 23 / 28 Theorem Let ϕ: Pd → R be a Minkowski linear functional. Then ϕ is positive, translation-invariant and symmetric if and only if there are real numbers c1,..., cd−1 ≥ 0 such that

d−1 X ϕ = ck fk . k=1

In particular, the family of all positive, Minkowski linear, translation- and symmetric functionals form a simplicial cone of dimension d − 1.

Generalized Permutohedra Symmetric Minkowski Linear Functionals

The Final step is to specialize the above theorem to the case when the functionals are also symmetric (under co-ordinate permutations).

For all 1 ≤ k ≤ d − 1 let fk : Pd → R be the symmetric, translation-invariant Minkowski linear functional deﬁned by i + 1d − i − 1 (f )(∆i+1) = (3.2) k 2 k − i for all 1 ≤ i ≤ d − 1.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 23 / 28 Generalized Permutohedra Symmetric Minkowski Linear Functionals

The Final step is to specialize the above theorem to the case when the functionals are also symmetric (under co-ordinate permutations).

Theorem Let ϕ: Pd → R be a Minkowski linear functional. Then ϕ is positive, translation-invariant and symmetric if and only if there are real numbers c1,..., cd−1 ≥ 0 such that

d−1 X ϕ = ck fk . k=1

In particular, the family of all positive, Minkowski linear, translation- and symmetric functionals form a simplicial cone of dimension d − 1.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 23 / 28 Generalized Permutohedra Completing the proof

The ﬁnal step is showing that the functional X X yI ∆I → yI h|I |, I ⊂[d] I ⊂[d] satisﬁes the above condition. This uses some basic combinatorics with univariate polynomials.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 24 / 28 Generalized Permutohedra Solid Angle Polynomials

d d Let q ∈ R be a point, P ⊆ R be a polytope and let B(q) denote the ball with radius centered at q. The solid angle of q with respect to P is deﬁned by

(P ∩ B(q)) ωq(P) = lim . →0 B

We note that the function q 7→ ωq(P) is constant on relative interiors of the faces of P. In particular, if q 6∈ P then ωq(P) = 0, if q is in the interior of P then ωq(P) = 1 and if q lies inside 1 the relative interior of a facet then ωq(P) = 2 . The solid angle sum of P is deﬁned by X A(P) = ωq(P) Fact : McMullen’s theorem shows this is a polynomial d q∈Z

Proposition 4 There is a 3-dimensional generalized permutahedron in R such that the linear term of its solid angle polynomial is negative.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 25 / 28 Perhaps the most basic problem in the theory of Ehrhart positivity is

Conjecture (Folklore) The Birkhoﬀ Polytope is Ehrhart positive.

This has been experimentally veriﬁed for n ≤ 11.

The Birkhoﬀ polytope is the Bipartite Matching polytope of the complete graph Kn,n. With very little evidence, we rashly conjecture

Conjecture Bipartite Matching polytopes are Ehrhart positive.

What about general matching polytopes? We conjecture

Conjecture General matching polytopes need not be Ehrhart positive.

Open Problems Open Problems

Definition (Birkhoff Polytope) The Birkhoff polytope is the set of all n × n matrices with non-negative entries so that each row and column sum is 1. This is a polytope of dimension (n − 1)2.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 26 / 28 The Birkhoﬀ polytope is the Bipartite Matching polytope of the complete graph Kn,n. With very little evidence, we rashly conjecture

Conjecture Bipartite Matching polytopes are Ehrhart positive.

What about general matching polytopes? We conjecture

Conjecture General matching polytopes need not be Ehrhart positive.

Open Problems Open Problems

Definition (Birkhoff Polytope) The Birkhoff polytope is the set of all n × n matrices with non-negative entries so that each row and column sum is 1. This is a polytope of dimension (n − 1)2.

Perhaps the most basic problem in the theory of Ehrhart positivity is

Conjecture (Folklore) The Birkhoﬀ Polytope is Ehrhart positive.

This has been experimentally veriﬁed for n ≤ 11.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 26 / 28 What about general matching polytopes? We conjecture

Conjecture General matching polytopes need not be Ehrhart positive.

Open Problems Open Problems

Definition (Birkhoff Polytope) The Birkhoff polytope is the set of all n × n matrices with non-negative entries so that each row and column sum is 1. This is a polytope of dimension (n − 1)2.

Perhaps the most basic problem in the theory of Ehrhart positivity is

Conjecture (Folklore) The Birkhoﬀ Polytope is Ehrhart positive.

This has been experimentally veriﬁed for n ≤ 11.

The Birkhoﬀ polytope is the Bipartite Matching polytope of the complete graph Kn,n. With very little evidence, we rashly conjecture

Conjecture Bipartite Matching polytopes are Ehrhart positive.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 26 / 28 Open Problems Open Problems

Definition (Birkhoff Polytope) The Birkhoff polytope is the set of all n × n matrices with non-negative entries so that each row and column sum is 1. This is a polytope of dimension (n − 1)2.

Perhaps the most basic problem in the theory of Ehrhart positivity is

Conjecture (Folklore) The Birkhoﬀ Polytope is Ehrhart positive.

This has been experimentally veriﬁed for n ≤ 11.

The Birkhoﬀ polytope is the Bipartite Matching polytope of the complete graph Kn,n. With very little evidence, we rashly conjecture

Conjecture Bipartite Matching polytopes are Ehrhart positive.

What about general matching polytopes? We conjecture

Conjecture General matching polytopes need not be Ehrhart positive.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 26 / 28 Conjecture The linear term of the Ehrhart polynomials on matroid polytopes is monotone with respect to deletion and contraction. We have veriﬁed this in over a hundred special cases.

Open Problems Monotonicity

Coming to matroid polytopes, consider the following operations on a matroid M = (E, T ). Deletion: Given e ∈ E, the matroid M \ e is the matroid on the ground set E \ e with independent sets being independent sets in M not containing e. Contraction: Given e ∈ E, the matroid M/e is the matroid on the ground set E \ e with independent sets being those sets such that appending e gives an independent set in M.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 27 / 28 Open Problems Monotonicity

Conjecture The linear term of the Ehrhart polynomials on matroid polytopes is monotone with respect to deletion and contraction. We have veriﬁed this in over a hundred special cases.

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 27 / 28 Open Problems Final Slide

Thanks for Listening!

Mohan Ravichandran, Bogazici, Istanbul Generalized Permutohedra : Ehrhart Positivity and Minkowski Linear Functionals 28 / 28