Volume of alcoved polyhedra and Mahler conjecture M.J. de la Puente - PowerPoint PPT Presentation

P ⊂ R n centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol( P ) increases, then vol( P ◦ ) decreases How large is vol( P ) vol( P ◦ )? Unit cube C n (edge= 2), C ◦ n is unit cross–polytope Unit ball B n = B ◦ n Blaschke–Santal´ o inequality (proved 1949): vol( P ) vol( P ◦ ) ≤ vol( B n ) vol( B ◦ n ) Mahler conjecture (open since 1939): vol( C n ) vol( C ◦ n ) ≤ vol( P ) vol( P ◦ ) Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

P ⊂ R n centrally symmetric convex body Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

P ⊂ R n centrally symmetric convex body Unit cube vol( C n ) = 2 n , Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

P ⊂ R n centrally symmetric convex body vol( C ◦ n ) = 2 n Unit cube vol( C n ) = 2 n , n ! Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

P ⊂ R n centrally symmetric convex body vol( C ◦ n ) = 2 n Unit cube vol( C n ) = 2 n , n ! π n / 2 Unit ball vol( B n ) = Γ( n 2 +1) Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

P ⊂ R n centrally symmetric convex body vol( C ◦ n ) = 2 n Unit cube vol( C n ) = 2 n , n ! π n / 2 Unit ball vol( B n ) = Γ( n 2 +1) 4 n π n ? ≤ vol( P ) vol( P ◦ ) ≤ Γ( n n ! 2 + 1) 2 Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

P ⊂ R n centrally symmetric convex body vol( C ◦ n ) = 2 n Unit cube vol( C n ) = 2 n , n ! π n / 2 Unit ball vol( B n ) = Γ( n 2 +1) 4 n π n ? ≤ vol( P ) vol( P ◦ ) ≤ Γ( n n ! 2 + 1) 2 is Mahler conj. Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

P ⊂ R n centrally symmetric convex body vol( C ◦ n ) = 2 n Unit cube vol( C n ) = 2 n , n ! π n / 2 Unit ball vol( B n ) = Γ( n 2 +1) 4 n π n ? ≤ vol( P ) vol( P ◦ ) ≤ Γ( n n ! 2 + 1) 2 is Mahler conj. and Blaschke–Santal´ o thm. Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

To compute the volume of a polytope Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 6/22

To compute the volume of a polytope is a # P –problem Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 6/22

To compute the volume of a polytope is a # P –problem (Dyer and Frieze 1988, L. Khachiyan 1989) Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 6/22

Alcoved polytope: Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Alcoved polytope: equations of facets of P ⊂ R n are of TWO types: Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Alcoved polytope: equations of facets of P ⊂ R n are of TWO types: x i = a i Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Alcoved polytope: equations of facets of P ⊂ R n are of TWO types: x i = a i x i − x j = a ij , Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Alcoved polytope: equations of facets of P ⊂ R n are of TWO types: x i = a i x i − x j = a ij , with a i , a ij ∈ R Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Alcoved polytope: equations of facets of P ⊂ R n are of TWO types: x i = a i x i − x j = a ij , with a i , a ij ∈ R ⇒ arrange coeffs. in matrix ( n + 1) × ( n + 1) ⇒ do linear algebra Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Alcoved polytope: equations of facets of P ⊂ R n are of TWO types: x i = a i x i − x j = a ij , with a i , a ij ∈ R ⇒ arrange coeffs. in matrix ( n + 1) × ( n + 1) ⇒ do linear algebra Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Alcoved polytope: equations of facets of P ⊂ R n are of TWO types: x i = a i x i − x j = a ij , with a i , a ij ∈ R ⇒ arrange coeffs. in matrix ( n + 1) × ( n + 1) ⇒ do linear algebra   0 − 6 − 6 A ( P ) = − 5 0 − 4   − 4 − 3 0 Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Alcoved polytope: equations of facets of P ⊂ R n are of TWO types: x i = a i x i − x j = a ij , with a i , a ij ∈ R ⇒ arrange coeffs. in matrix ( n + 1) × ( n + 1) ⇒ do linear algebra   0 − 6 − 6 A ( P ) = − 5 0 − 4   − 4 − 3 0 a i , n +1 ≤ x i ≤ − a n +1 , i Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Alcoved polytope: equations of facets of P ⊂ R n are of TWO types: x i = a i x i − x j = a ij , with a i , a ij ∈ R ⇒ arrange coeffs. in matrix ( n + 1) × ( n + 1) ⇒ do linear algebra   0 − 6 − 6 A ( P ) = − 5 0 − 4   − 4 − 3 0 a i , n +1 ≤ x i ≤ − a n +1 , i a i , j ≤ x i − x j ≤ − a j , i Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Alcoved polytope: equations of facets of P ⊂ R n are of TWO types: x i = a i x i − x j = a ij , with a i , a ij ∈ R ⇒ arrange coeffs. in matrix ( n + 1) × ( n + 1) ⇒ do linear algebra   0 − 6 − 6 A ( P ) = − 5 0 − 4   − 4 − 3 0 a i , n +1 ≤ x i ≤ − a n +1 , i a i , j ≤ x i − x j ≤ − a j , i a ii = 0 Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

Which matrices [ a ij ] ∈ M n +1 yield alcoved polytopes P ⊂ R n ? Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

Which matrices [ a ij ] ∈ M n +1 yield alcoved polytopes P ⊂ R n ? Thm. (Sergeev 2009, de la Puente 2013) Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

Which matrices [ a ij ] ∈ M n +1 yield alcoved polytopes P ⊂ R n ? Thm. (Sergeev 2009, de la Puente 2013) Every normal idempotent matrix gives rise to an alcoved polytope Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

Which matrices [ a ij ] ∈ M n +1 yield alcoved polytopes P ⊂ R n ? Thm. (Sergeev 2009, de la Puente 2013) Every normal idempotent matrix gives rise to an alcoved polytope A = [ a ij ] normal matrix: a ii = 0 and a ij ≤ 0, all i , j Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

Which matrices [ a ij ] ∈ M n +1 yield alcoved polytopes P ⊂ R n ? Thm. (Sergeev 2009, de la Puente 2013) Every normal idempotent matrix gives rise to an alcoved polytope A = [ a ij ] normal matrix: a ii = 0 and a ij ≤ 0, all i , j A idempotent (wrt tropical product): A ⊙ A = A Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

Which matrices [ a ij ] ∈ M n +1 yield alcoved polytopes P ⊂ R n ? Thm. (Sergeev 2009, de la Puente 2013) Every normal idempotent matrix gives rise to an alcoved polytope A = [ a ij ] normal matrix: a ii = 0 and a ij ≤ 0, all i , j A idempotent (wrt tropical product): A ⊙ A = A ⊕ = max Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

Which matrices [ a ij ] ∈ M n +1 yield alcoved polytopes P ⊂ R n ? Thm. (Sergeev 2009, de la Puente 2013) Every normal idempotent matrix gives rise to an alcoved polytope A = [ a ij ] normal matrix: a ii = 0 and a ij ≤ 0, all i , j A idempotent (wrt tropical product): A ⊙ A = A ⊕ = max is trop. sum Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

Which matrices [ a ij ] ∈ M n +1 yield alcoved polytopes P ⊂ R n ? Thm. (Sergeev 2009, de la Puente 2013) Every normal idempotent matrix gives rise to an alcoved polytope A = [ a ij ] normal matrix: a ii = 0 and a ij ≤ 0, all i , j A idempotent (wrt tropical product): A ⊙ A = A ⊕ = max is trop. sum and ⊙ = + Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

Which matrices [ a ij ] ∈ M n +1 yield alcoved polytopes P ⊂ R n ? Thm. (Sergeev 2009, de la Puente 2013) Every normal idempotent matrix gives rise to an alcoved polytope A = [ a ij ] normal matrix: a ii = 0 and a ij ≤ 0, all i , j A idempotent (wrt tropical product): A ⊙ A = A ⊕ = max is trop. sum and ⊙ = + is trop. product Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

⊕ = max tropical sum , ⊙ = + trop. product Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

⊕ = max tropical sum , ⊙ = + trop. product     0 − 6 − 6 0 − 2 − 6 − 5 0 − 4 0 0 0 A = A ⊕ B =     − 4 − 3 0 − 4 − 3 0 A ⊙ B =   0 − 2 − 7   0 − 2 − 6 B = 0 0 0   max {− 5 , 0 , − 9 } 0 0 − 5 − 5 0   max {− 4 , − 3 , − 5 } − 3 0 A , B are normal A idempotent , Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

⊕ = max tropical sum , ⊙ = + trop. product     0 − 6 − 6 0 − 2 − 6 − 5 0 − 4 0 0 0 A = A ⊕ B =     − 4 − 3 0 − 4 − 3 0 A ⊙ B =   0 − 2 − 7   0 − 2 − 6 B = 0 0 0   max {− 5 , 0 , − 9 } 0 0 − 5 − 5 0   max {− 4 , − 3 , − 5 } − 3 0 A , B are normal   0 − 2 − 2 A idempotent , B ⊙ B = 0 0 0  ,  − 5 − 5 0 Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

⊕ = max tropical sum , ⊙ = + trop. product     0 − 6 − 6 0 − 2 − 6 − 5 0 − 4 0 0 0 A = A ⊕ B =     − 4 − 3 0 − 4 − 3 0 A ⊙ B =   0 − 2 − 7   0 − 2 − 6 B = 0 0 0   max {− 5 , 0 , − 9 } 0 0 − 5 − 5 0   max {− 4 , − 3 , − 5 } − 3 0 A , B are normal   0 − 2 − 2 A idempotent , B ⊙ B = 0 0 0  , B not  − 5 − 5 0 idempotent Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

⊕ = max tropical sum , ⊙ = + trop. product     0 − 6 − 6 0 − 2 − 6 − 5 0 − 4 0 0 0 A = A ⊕ B =     − 4 − 3 0 − 4 − 3 0 A ⊙ B =   0 − 2 − 7   0 − 2 − 6 B = 0 0 0   max {− 5 , 0 , − 9 } 0 0 − 5 − 5 0   max {− 4 , − 3 , − 5 } − 3 0 A , B are normal   0 − 2 − 2 A idempotent , B ⊙ B = 0 0 0  , B not  − 5 − 5 0 idempotent P ( A ) alcoved , Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

⊕ = max tropical sum , ⊙ = + trop. product     0 − 6 − 6 0 − 2 − 6 − 5 0 − 4 0 0 0 A = A ⊕ B =     − 4 − 3 0 − 4 − 3 0 A ⊙ B =   0 − 2 − 7   0 − 2 − 6 B = 0 0 0   max {− 5 , 0 , − 9 } 0 0 − 5 − 5 0   max {− 4 , − 3 , − 5 } − 3 0 A , B are normal   0 − 2 − 2 A idempotent , B ⊙ B = 0 0 0  , B not  − 5 − 5 0 idempotent P ( A ) alcoved , P ( B ) not alcoved Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

P ( A ) alcoved (hence convex ), P ( B ) not alcoved Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 10/22

dodecahedron with f –vector ( v , e , f ) = (20 , 30 , 12) Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 11/22

For n = 4 get volume of alcoved P ⊂ R 3 Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? normal idempotent A = B − E Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? normal idempotent A = B − E define bounding box B Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? normal idempotent A = B − E define bounding box B def. six cants Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? normal idempotent A = B − E define bounding box B def. six cants arranged in cycle Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? normal idempotent A = B − E define bounding box B def. six cants arranged in cycle Easy: vol( P ) = vol(box) − � 6 j =1 vol P j + � j vol( P j ∩ P j +1 ) Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? normal idempotent A = B − E define bounding box B def. six cants arranged in cycle Easy: vol( P ) = vol(box) − � 6 j =1 vol P j + � j vol( P j ∩ P j +1 ) Thm.: Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? normal idempotent A = B − E define bounding box B def. six cants arranged in cycle Easy: vol( P ) = vol(box) − � 6 j =1 vol P j + � j vol( P j ∩ P j +1 ) 6 6 c 2 m 2 − m 3 j ℓ j j M j � � j Thm.: vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? normal idempotent A = B − E define bounding box B def. six cants arranged in cycle Easy: vol( P ) = vol(box) − � 6 j =1 vol P j + � j vol( P j ∩ P j +1 ) 6 6 c 2 m 2 − m 3 j ℓ j j M j � � j Thm.: vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ 1 , ℓ 2 , ℓ 3 are edge–lengths of bounding box B Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? normal idempotent A = B − E define bounding box B def. six cants arranged in cycle Easy: vol( P ) = vol(box) − � 6 j =1 vol P j + � j vol( P j ∩ P j +1 ) 6 6 c 2 m 2 − m 3 j ℓ j j M j � � j Thm.: vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ 1 , ℓ 2 , ℓ 3 are edge–lengths of bounding box B m j := min {| c j | , | c j +1 |} , Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

For n = 4 get volume of alcoved P ⊂ R 3 in terms of matrix entries a ij of A ∈ M 4 ? normal idempotent A = B − E define bounding box B def. six cants arranged in cycle Easy: vol( P ) = vol(box) − � 6 j =1 vol P j + � j vol( P j ∩ P j +1 ) 6 6 c 2 m 2 − m 3 j ℓ j j M j � � j Thm.: vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ 1 , ℓ 2 , ℓ 3 are edge–lengths of bounding box B m j := min {| c j | , | c j +1 |} , M j := max {| c j | , | c j +1 |} , c j is cant parameter , j = 1 , 2 , . . . , 6 Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 13/22

Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 13/22

Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 13/22

6 6 c 2 m 2 − m 3 j ℓ j j M j j � � vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

6 6 c 2 m 2 − m 3 j ℓ j j M j j � � vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ k = a k 4 , k = 1 , 2 , 3 Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

6 6 c 2 m 2 − m 3 j ℓ j j M j j � � vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ k = a k 4 , k = 1 , 2 , 3 each c j cant parameter is the difference of two entries in A Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

6 6 c 2 m 2 − m 3 j ℓ j j M j j � � vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ k = a k 4 , k = 1 , 2 , 3 each c j cant parameter is the difference of two entries in A m j := min {| c j | , | c j +1 |} , Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

6 6 c 2 m 2 − m 3 j ℓ j j M j j � � vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ k = a k 4 , k = 1 , 2 , 3 each c j cant parameter is the difference of two entries in A m j := min {| c j | , | c j +1 |} , M j := max {| c j | , | c j +1 |} , Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

6 6 c 2 m 2 − m 3 j ℓ j j M j j � � vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ k = a k 4 , k = 1 , 2 , 3 each c j cant parameter is the difference of two entries in A m j := min {| c j | , | c j +1 |} , M j := max {| c j | , | c j +1 |} , vol( P ) is deg 3 rational homog polynomial in a ij Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

6 6 c 2 m 2 − m 3 j ℓ j j M j j � � vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ k = a k 4 , k = 1 , 2 , 3 each c j cant parameter is the difference of two entries in A m j := min {| c j | , | c j +1 |} , M j := max {| c j | , | c j +1 |} , vol( P ) is deg 3 rational homog polynomial in a ij A = B − E Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

6 6 c 2 m 2 − m 3 j ℓ j j M j j � � vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ k = a k 4 , k = 1 , 2 , 3 each c j cant parameter is the difference of two entries in A m j := min {| c j | , | c j +1 |} , M j := max {| c j | , | c j +1 |} , vol( P ) is deg 3 rational homog polynomial in a ij A = B − E ℓ k are the dimensions of bounding box B Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

6 6 c 2 m 2 − m 3 j ℓ j j M j j � � vol( P ) = ℓ 1 ℓ 2 ℓ 3 − + 2 2 6 j =1 j =1 ℓ k = a k 4 , k = 1 , 2 , 3 each c j cant parameter is the difference of two entries in A m j := min {| c j | , | c j +1 |} , M j := max {| c j | , | c j +1 |} , vol( P ) is deg 3 rational homog polynomial in a ij A = B − E ℓ k are the dimensions of bounding box B the entries of E are the c j ’s or zero Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

Mahler conjecture for 3–dim alcoved polytopes Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

Mahler conjecture for 3–dim alcoved polytopes Thm. (de la Puente, 2012) Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

Mahler conjecture for 3–dim alcoved polytopes Thm. (de la Puente, 2012) A symmetric matrix ⇔ P centrally symmetric Vol. alc. polyhedr. Mahler conj. M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22