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 2 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 2 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 2 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 . A map φ : Kd ! R is called a valuation if for every A; B 2 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. 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 . A map φ : Kd ! R is called a valuation if for every A; B 2 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. 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 . A map φ : Kd ! R is called a valuation if for every A; B 2 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. 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 . A map φ : Kd ! R is called a valuation if for every A; B 2 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. 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 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. 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 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. 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.