Volume of Alcoved Polyhedra and Mahler Conjecture

María J. de la Puente Pedro L. Clavería Universidad Complutense Universidad de Zaragoza Facultad de Mátemáticas Zaragoza, Spain Madrid, Spain [email protected] [email protected]

ABSTRACT in the 1980’s. The difficulty is present, no matter whether the de- The facet equations of a 3–dimensional alcoved polyhedron P are scription is by given the list of vertices or the facet equations. The only of two types (xi = cnst and xi −xj = cnst) and the f –vector of Mahler conjecture (an easy–to–state question on the product of the P is bounded above by (20,30,12). In general, P is a dodecahedron volumes of a symmetric polytope and its polar) remains open since with 20 vertices and 30 edges. We represent an alcoved polyhedron 1938, although progress has been reported [11, 12, 16, 24]. by a real square matrix A of order 4 and we compute the exact Volumes can be used to compute intersection numbers in alge- volume of P: it is a polynomial expression in the aij , homogeneous braic geometry. More general than volume is the concept of mixed of degree 3 with rational coefficients. Then we compute the volume volume, introduced by H. Minkowski. In the survey paper [8], the of the polar P◦, when P is centrally symmetric. Last, we show that authors ask for families of polytopes for which volume and mixed Mahler conjecture holds in this case: the product of the volumes of volumes can be computed. We propose alcoved polytopes (as de- P and P◦ is no less that 43/3!, with equality only for boxes. Our scribed in this paper) for that role, as well as more general alcoved proof reduces to computing a certificate of non–negativeness ofa polytopes (as in [18, 19, 25]). Volumes of matroid polytopes are certain polynomial (in 3 variables, of degree 6, non homogeneous) introduced in [2]. A lack of concrete non–trivial worked examples on a certain simplex. in these papers is filled in by the present paper. A frequent problem in real algebraic geometry is checking whether CCS CONCEPTS a given polynomial is positive on a given semi–algebraic set. This and related questions were studied by D. Hilbert and H. Minkowski. • Computing methodologies → Representation of mathemat- The foundational statement is Hilbert 17th problem. An algebraic ical objects; method to give an answer to such queries is to show a certificate, KEYWORDS i.e., to express the given polynomial with a formula for which the positiveness is self–evident. volume, alcoved polyhedron, tropical semiring, idempotent semir- In classical linear algebra one works with matrices. A linear ing, dioid, normal matrix, idempotent matrix, symmetric matrix, map between finite–dimensional spaces is represented by a matrix, perturbation, Mahler conjecture, polynomial, certificate of non– after having fixed bases. Here we do something similar: an alcoved negativeness. polyhedra is represented by a matrix. Then we develop a dictionary: ACM Reference Format: properties of alcoved polyhedra are converted into properties of María J. de la Puente and Pedro L. Clavería. 2018. Volume of Alcoved Polyhe- matrices, and viceversa. Our matrix–based technique has a big dra and Mahler Conjecture. In ISSAC ’18: 2018 ACM International Symposium potential and could be extended to higher dimensions and to alcoved on Symbolic and Algebraic Computation, July 16–19, 2018, New York, NY, USA. based on different root systems. ACM, New York, NY, USA, 8 pages. https://doi.org/10.1145/3208976.3208990 Our main result is the exact volume formula in theorem 4.1. As application, we show that Mahler conjecture for alcoved centrally 1 INTRODUCTION symmetric polyhedra holds true. In this case, the conjecture is The theory of polytopes is a meeting point for optimization, convex equivalent to the following assertion: the polynomial MC displayed geometry, lattice geometry and geometry of numbers. There are in p. 7 is non–negative on a certain simplex S ⊆ R3. We show that deep results, computationally difficult problems and open conjec- this is true, by finding a certificate of non–negativeness of MC over tures concerning polytopes, even in three–dimensional space, as S, i.e., we express MC as a linear combination, with non–negative the collective book [22] beautifully shows. The computation of the coefficients, of products of polynomials w1,w2,w3,w4 such that wj volume is ♯-P, as M.E. Dyer and A.M. Frieze (together with Valiant’s is non–negative on S, all j. The wj provide equations for the facets result on permanents) and, independently, L. Khachiyan showed of S. We work with normal matrices operated tropically. We further Permission to make digital or hard copies of all or part of this work for personal or need to impose tropical idempotency on matrices in order to achieve classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation convexity (alcoved polytopes, in our setting). Without idempotency, on the first page. Copyrights for components of this work owned by others than ACM we do not get one convex set, but a complex of such sets. Among must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, NI matrices, we concentrate on two matrix families: visualized to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. and symmetric. Visualized matrices are easy to work with because ISSAC ’18, July 16–19, 2018, New York, NY, USA in them we clearly distinguish two parts: a box matrix and a per- © 2018 Association for Computing Machinery. turbation matrix. Accordingly, the alcoved polyhedron is simply ACM ISBN 978-1-4503-5550-6/18/07...$15.00 https://doi.org/10.1145/3208976.3208990 a perturbed box, where we are able to read off the edge–lengths. Symmetric matrices are useful because, as our dictionary shows, an unique way, using the Hungarian algorithm [17]). Normalization alcoved polyhedron is symmetric with respect to the origin if and consists of a translation and a rearrangement of columns. This is only if its defining matrix is symmetric. explained below. The oldest known volume formula is Egyptian, found in Problem Definition 2.2. In M , 14 in the Moscow Mathematical Papyrus (ca. 1850 BC). It expresses n the volume of a frustum of a square pyramid. Later, Piero della (1) a matrix D = [dij ] ∈ Mn is diagonal if dii is real and dij = Francesca (s. XVI) and Carnot (s. XVIII) found polynomial expres- −∞, if i , j. The inverse of the diagonal matrix D, denoted −1 sions with rational coefficients for the square of the volume ofa D , is [aij ] such that aii = −dii and aij = −∞, if i , j. σ tetrahedron in terms of the edge–lengths. Both expressions can be (2) For a permutation σ ∈ Sn, the permutation matrix P = [aij ] rewritten, obtaining the Cayley–Menger determinantal formula. is defined by aiσ (i) = 0, aij = −∞, otherwise. Our volume formula has the same flavor, for two reasons. First, Definition 2.3. For a matrix A ∈ Mn, we define it is a polynomial formula, with rational coeffcients. Second, any (1) row(A,j) to be the j–th row of A, and col(A,j) to be the j–th alcoved polyhedron is a tetrahedron, in the sense that it is spanned column of A, all j ∈ [n], (tropically) by four points. (2) if row(A,n) is real, then the visualization of A is the matrix Tropical mathematics (also called Idempotent mathematics or −1 A0 := AD , where D is the diagonal D = diag(row(A,n)). Max–plus mathematics) is the setting of this paper. There are several Obviously, row(A0,n) is all–zero. books and paper collections on the subject (with grown branches in analysis, geometry, algebra, etc.) For algebra, we refer the reader to Visualization has a meaning. Think of the columns of a matrix [1, 4, 20]. Normal matrices can be traced back to a paper of M. Yoeli A ∈ Mn as n points in n–dimensional space. Consider n parallel in the 1960’s (under a different name). Lately, they have been used lines passing through these points with directional vector (1,...,1). by Butcovič [4] and thoroughly studied in the thesis [21] (under a Intersect these n lines with the hyperplane {xn = 0} and write the different name). Idempotent tropical matrices have been usedin coordinates of the resulting points as the columns of a matrix. This [14]. Visualized tropical matrices have been used in [23]. matrix is A0. When the matrix is visualized, we have A = A0. Volumes for non–square classical (i.e., non–tropical) matrices What do we want to see? The set span A gathers of all tropical have been introduced in [3]. linear combinations of columns of A ∈ Mn span A := {λ col(A,1) ⊕ · · · ⊕ λ col(A,n) : λ ,...,λ ∈ R}. (1) 2 MATRICES WITH TROPICAL OPERATIONS 1 n 1 n Clearly span A = span A (because columns of A and A are pro- For n ∈ N, let [n] denote the set {1,2,...,n}. Consider the tropical 0 0 portional). semiring max–plus semiring (R, ⊕, ⊙) R = R ∪ (also called ) , where : Normality has a geometric meaning: A is normal if and only if {−∞} is the extended real numbers and a ⊕ b = max{a,b} and a ⊙ b = a + b, a,b ∈ R. The neutral element with respect to tropical col(A0,j) ∈ Rj , all j ∈ [n]. (2) addition ⊕ is −∞, and the neutral element with respect to tropical where the regions multiplication ⊙ is 0. Addition is idempotent, because a ⊕ a = R = {x ∈ Rn−1 ≤ x x ≤ x , k ∈ n − }, j ∈ n − , max{a,a} = a, so that (R, ⊕, ⊙) is an idempotent semiring or dioid. j : : 0 j and k j [ 1] [ 1] n−1 Let Mn be the set of order n square matrices over R. The tropical Rn := {x ∈ R : xk ≤ 0, k ∈ [n − 1]}. operations are extended to matrices in the standard way. (Mn, ⊕, ⊙) n−1 n n−1 provide a closed covering of R = ∪ Rj and R is identified is a semiring. We also add matrices classically, but we never multiply j=1 with {x = 0} inside Rn (see figure 1). them classically. Therefore, we omit the symbol ⊙ between matrices n (for simplicity). For example: 1 2 −1 0 0 2 A = , B = , A + B = " 3 4 # " −∞ 5 # " −∞ 9 # 1 2 max{0,−∞} max{1,7} 0 7 A⊕B = ,AB = = . " 3 5 # " max{2,−∞} max{3,9} # " 2 9 #

Definition 2.1. A matrix A = [aij ] ∈ Mn is normal if aii = 0 and aij ≤ 0, all i,j ∈ [n]. Figure 1: Coordinate axes in the plane (left), closed regions Why normal? The matrix [a ] ∈ M with a = 0 and a = −∞, ij n ii ij R ,R ,R in the plane (center) and coordinate axes in 3– for i j is denoted I . The all–zero matrix is denoted Z . 1 2 3 , n n dimensional space (right). A normal matrix A satisfies AZn = Zn = ZnA (but a general matrix does not). A matrix A is normal if and only if In ≤ A ≤ Zn if and only if I ≤ A ≤ A2 ≤ A3 ≤ · · · ≤ Z (since matrix tropical n n Definition 2.4. A matrix A ∈ Mn is visualized if A = A0. multiplication is monotonic). N Definition 2.5. Given matrices A,D,Pσ ∈ M with D diagonal In the set Mn of normal matrices, notice that In is the identity n N and Pσ permutation matrix, for both tropical addition and multiplication in Mn (but not in Mn). Restricting to work with normal matrices is advantageous and no (1) the translate of A by D is the matrix DA, serious limitation. Indeed, every matrix can be normalized (in a non (2) the conjugate of A by D is the matrix DA := DAD−1, (3) the matrix APσ is the result of relabeling the columns of A according to the permutation σ.

In the paragraph prior to definition 2.2, we said that every ma- trix M ∈ Mn can be normalized. This means that a there exists a translate DM of M, and a relabeling of columns DMP of DM, such that A = DMP satisfies (2).

2 Definition 2.6. A matrix A = [aij ] ∈ Mn is idempotent if A = A. Thus, a matrix A is normal idempotent if and only if I ≤ A2 ≤ n Figure 2: Cantable edges (in dashed blue lines) in a cube A ≤ Zn if and only if make a cycle: ℓ1 is front top, ℓ2 is top left, ℓ3 is left back, ℓ4 is = + ≤ ≤ , , , ∈ . aii 0 and aij ajk aik 0 all i j k [n] (3) back bottom, ℓ5 is bottom right and ℓ6 is right front. Genera- NI tors are marked with big blue dots and labeled 1,2,3,4, indi- Let Mn the set of normal idempotent matrices of order n. cating the column in A each of them is: 1 is bottom right, 2 D 0 Corollary 2.7. Any conjugate A is normal (resp. idempotent), is bottom back, 3 is top left, 4 is bottom left. whenever A is. T T A matrix is symmetric if A = A , where A is the classical trans- satisfies O = maxv (P). This is easily achieved matrix–wise, as SNI P posed matrix. Let Mn the set of symmetric normal idempotent the next lemma shows. matrices and MVNI be the set of visualized normal idempotent n NI matrices. The only symmetric and visualized normal idempotent Lemma 3.2. For A ∈ Mn , the matrix A is visualized if and only if O = max P(A). matrix is the zero matrix Zn. In the rest of paper we work with normal idempotent matrices, Proof. We know that P(A) is alcoved. Assume A = A0 is NI. By the reason being that they represent alcoved polyhedra. (2) we have col(A0,j) ∈ Rj ∩ Rn, all j, whence O ∈ P(A) and P(A) is contained in the non–positive octant Rn. Thus O = max P(A). 3 ALCOVED POLYHEDRA FROM NORMAL The converse is similar.  IDEMPOTENT MATRICES Mahler conjecture deals with centrally symmetric bodies. For In order to study the set span A ⊂ Rn defined in (1), it is enough a centrally symmetric alcoved polytope P ⊂ Rn−1, a preferred to study its intersection with the hyperplane {xn = 0} (identified n−1 n−1 translation is sP : R → R such that O is the center of with Rn−1), because both sets determine each other symmetry of sP (P). This is easily achieved matrix–wise, as the P(A) := span A ∩ {xn = 0}. (4) next lemma shows. For a general matrix A ∈ M , the set P(A) ⊂ Rn−1 is a polytopal NI n Lemma 3.3 (Lemma 3 in [13]). For A ∈ Mn , the matrix A is complex of impure dimension no bigger that n − 1. However, if symmetric if and only if −P(A) = P(A) (i.e., O is the center of A is normal idempotent, then P(A) reduces to just one convex symmetry of P(A)). polytope and this polytope is alcoved (see [5, 6, 13, 15, 18, 19, ∈ R4 23, 25]). In other words, normality and idempotency prevents the Example 3.4. Given a column vector t ≤0 with coordinates T existence of lower dimensional parts in P(A). (t1,t2,t3,0) , the following matrix (easily checked to be idempotent) is called visualized box matrix Definition 3.1. An alcoved polytope in Rn−1 is a bounded poly- 0 t t t tope P whose facet equations are of type x = cnst, and x − x = 1 1 1 i i j t 0 t t cnst, i,j ∈ [n − 1].A box in Rn−1 is a bounded polytope B whose VB(t) :=  2 2 2  ∈ MVNI . (5)  t t 0 t  4 facet equations are of type x = cnst, i ∈ [n − 1].A cube is a box  3 3 3  i  0 0 0 0  of equal edge–lengths.   A conjugate of VB(t) is the matrix SB(t) = DVB(t), with D = Clearly, the property of being alcoved is preserved by translation.   diag(−t/2)). We have SB(t) = (ttT )/2 ⊕ I = [b ], with b = 0 The simplest alcoved polytopes are boxes. Every alcoved polytope 4 ij ii and b = (t +t )/2, if i j. The matrix SB(t) is called symmetric is a perturbed box, more precisely, it is a canted box. The verb to ij i j , box matrix and we easily get cant means to bevel, to form an oblique surface upon something. We cant edges of boxes, always at an angle of π/4 radians (45 degrees). −t1/2 t1/2 t1/2 t1/2 To cant an edge in a box means to create a new facet. 45 degrees t2/2 −t2/2 t2/2 t2/2 SB(t)0 =   . (6) is the angle determined by the intersecting planes xi = cnst and  t3/2 t3/2 −t3/2 t3/2  − 3  0 0 0 0  xi xj = cnst. Note that Assume n = 4, so our box is in R . Only   six edges of a box are cantable: front top, top left, left back, back  3  The polyhedron P (SB(t)) is a box in R , centered at O, whose bottom, bottom right and right front. Note that the cantable edges   edge–lengths are |t1|, |t2|, |t3|. The polyhedron P(VB(t)) is a trans- are arranged in a cycle: ℓ1,ℓ2,. . . ,ℓ6 (see figure 2). late of P(SB(t)), satisfyingO = max P(VB(t)). Note that min P(VB(t)) For each alcoved polytope P ⊂ Rn−1 we have a preferred trans- is the vector t. n−1 n−1 n−1 lation; it is vP : R → R such that the origin O of R These boxes are cubes if ti = tj , i , j. We will work in R3, from now on. Definition 3.5 (Perturbation). For a visualized normal idempotent ∈ VNI matrix V = [vij ] M4 , we write B := VB(col(V ,4)) (as in (5)) and E := B − V . The matrix E is called perturbation matrix of V . By definition, B is a visualized matrix box. We say that V is the result of perturbing B by E. Similarly, we say that P(B) is the bounding box of P(V ) (see figure 3). Notice that a perturbation matrix E is normal visualized, but not idempotent, in general. Figure 4: Two prisms P1 (left) and P6 (right) to be intersected.

Figure 5: The wedge P6 ∩ P1.

are organized in a cycle. The intersection of two consecutive prisms Figure 3: One cant performed on a box. The box is contained has been removed twice, so it must be added once. We get in the non–positive orthant R3. The origin O is marked in 6 red. Generators, i.e., columns of the defining matrix A = A0 X X are marked in blue big dots and labeled 1,2,3,4. We have O = vol P(V ) = vol P(B)− vol Pj + vol(Pj ∩Pj+1). (8) P − j=1 j ∈[6] mod 6 max (A) with A = B √E and eij = 0 unless (i,j) = (2,3). Notice 2 the edge–length |e23| 2. cj ℓj Clearly we have vol Pj = 2 , where ℓi = ℓi−3, i = 4,5,6. In addition, the intersection Pj ∩ Pj+1 is a wedge, (i.e., a prism Pj T Definition 3.6 (Cant tuple). For V = B − E ∈ MVNI as in defini- from which a tetrahedron has been removed) (see figures 4 and 4 m2M m3 ∈ 6 (P ∩ P ) = P − T = j j − j tion 3.5, the cant tuple of V is c = (cj ) R≤0, with c1 := e23, 5). Thus vol j j+1 vol j vol 2 6 .  c2 := e13, c3 := e12, c4 := e32, c5 := e31 and c6 := e21. Set R3 mj := min{|cj |, |cj+1|}, Mj := max{|cj |, |cj+1|}, j ∈ [6] and c7 = c1. After a symmetry of (i.e., a map preserving distances and P Also set ℓi := |vi4|, i ∈ [3]. angles), each prism j is an alcoved polyhedron. The same holds for the tetrahedron T appearing in the former proof. Tetrahedra as In summary, every alcoved polytope is a perturbed box. A box is such have been studied by M. Fiedler, and called Schläfli simplex by rather degenerate (i.e., non–maximal) alcoved polyhedron. this author. T is a right simplex whose tree of legs is a path (see [7]). 4 VOLUME OF AN ALCOVED POLYHEDRON Example 4.2. We want to compute the volume of the polyhe- In this section we show that the volume of an alcoved convex dron P defined by the inequalities −7 ≤ x1 ≤ 1, −6 ≤ x2 ≤ 1, polyhedron P is a cubic homogeneous polynomial in the entries vij −5 ≤ x ≤ −3, −6 ≤ x −x ≤ 1, −8 ≤ x −x ≤ 3, −1 ≤ x −x ≤ 8. of a defining matrix V for P, with rational coefficients. Multiplying 3 1 2 2 3 3 1 0 −6 −8 −7 by 3! we get integral coefficients. −1 0 −8 −6 The volume is a valuation, i.e., vol(P ) + vol(P ) = vol(P ∪ A defining matrix is A =   , which is normal 1 2 1  −1 −3 0 −5  P ) + vol(P ∩ P ). Further, the volume of P is preserved under   1 1 2  −1 −2 −3 0  translation, so we can assume that a defining matrix for P is VNI.   and idempotent. The conjugate matrix of A by D= diag(row(A,4)) ∈ VNI 0 −5 −6 −8  Theorem 4.1. For V = [vij ] M4 , take cj ,mj ,Mj ,ℓj as in P −2 0 −7 −8 definition 3.6. Then the volume of the alcoved polytope (V ) is is V = V =   and P(V ) is a translate of P. 0  −3 −4 0 −8  6 m2M m3 c2ℓ   X j j j j j  0 0 0 0  vol P(V ) = ℓ1ℓ2ℓ3 + − − . (7)   2 6 2 V  V= B − E j=1 To we apply theorem 4.1: we let  with perturbation ma- 0 −3 −2 0  Proof. Write V = B − E. The volume of the bounding box is −6 0 −1 0 trix E =   and cubic box of edge–length 8. The ℓ1ℓ2ℓ3. From this box we remove six right prisms Pj , j ∈ [6]. The  −5 −4 0 0  base of prism P is a right isosceles triangle legged |c |. The prisms  0 0 0 0  j j       Lemma 5.2 (Entries of conjugate matrix). Let V = DA = ∈ VNI ∈ NI [vij ] M4 be the conjugate of A = [aij ] M4 by D = diag(row(A,4)) and write V = B − E, with perturbation matrix E = [eij ]. Then

vij = a4i;ij , eij = a4i;j4. (11) 6 SYMMETRY In this section we show that symmetry is transferred from a matrix S to the perturbation matrix E of its conjugate DS, and conversely. ∈ SNI Lemma 6.1 (Conjugation and symmetry). (1) If S M4 andV = DS = B−E is the conjugate of S by D = diag(row(S,4)), then E is symmetric. − ∈ VNI D ∈ SNI (2) If V = B E M4 with E symmetric, then V M4 , with D = diag(−v14/2,−v24/2,−v34/2,0). (3) In either case if, in addition, B is a visualized cube matrix, then sij = vij , for i,j ∈ [3]. Figure 6: The polyhedron P in example 4.2. It is a dodeca- hedron. The generators, i.e., the columns of matrix A0 are Proof. From lemma 5.2, S = ST and equalities (10), we get marked with big blue dots and labeled 1,2,3,4. T eij = s4i;j4 = sj4;4i = s4j;i4 = eji , whence E = E , proving item 1. D ′ For the proof of item 2 write S = V . We have sij = vij;j4 if j ∈ [3], s = v′ , where V ′ = [v′ ] is an auxiliary matrix such that cant tuple is c = (−1,−2,−3,−4,−5,−6), whence m = (1,2,3,4,5,1), i4 i4 ij ′ / ′ T M = (2,3,4,5,6,6), ℓj = 8, so that vi4 = vi4 2 and vij = vij , otherwise. Then S = S follows from E = ET S V S 6 2 6 3 6 2 , idempotent follows from idempotent and normal X mj Mj 428 X mj 113 X cj ℓj T = , = , = 364 follows from S = S and V normal. 2 3 6 3 2 ∈ j= j= j= Last, if B is a visualized cube matrix, then vi4 = vj4, for i,j [3], 1 1 1 ′ whence sij = v = vij + vj4/2 − vi4/2 − vjj = vij , proving item 428 113 760 ij;j4 giving a total value vol P = vol P(V ) = 512+ 3 − 3 −364 = 3 . 3.  Note that the former polyhedron P is maximal, i.e., its f –vector Corollary 6.2 (Volume formula for V such that E = ET ). is (f , f , f ) = (20,30,12) (see [5, 6, 13, 15]) and P is a dodecahe- 0 1 2 If V = B − E ∈ MVNI with symmetric E, then dron. For general n, the number f of vertices is bounded above by 4   0 2n−2 , as proved in [6]. 3 m3 n−1 X 2 j 2 vol P(V ) = ℓ ℓ ℓ + m Mj − − c ℓj . (12) 1 2 3 j 3 j 5 THE 2–MINORS OF A MATRIX j=1 How do we compute the edge–lengths of the alcoved P(A) from the Proof. We have cj = cj+3, mj = mj+3 and Mj = Mj+3.  entries aij of A? The answer is easy if the matrix A is VNI: take the bij = ai4 and the eij , for A = B − E, B = [bij ] box matrix, E = [eij ] 7 POLARS AND MAHLER CONJECTURE perturbation matrix (see figure 3). In general, if A is only NI, the Let r ∈ N and p ,p ,...,p be vectors in Rn. Let ⟨,⟩ denote the first thing we must do is computing the conjugate matrix V = DA. 1 2 r standard inner product. If P = {x ∈ Rn : ⟨x,p ⟩ ≤ 1,∀k ∈ [r]}, In this section we introduce the 2–minors of A and show that k then the polar P◦ is defined as conv(p ,p ,...,p ), the convex hull the entries of the conjugate matrix V are certain 2–minors of A. 1 2 r of vectors p ,p ,...,p ; see figure 7. Moreover, if O belongs to the The same is true for the perturbation matrix E of V . 1 2 r interior of P, then (P◦)◦ = P. a b Let (v ,v ,v ) be the canonical basis in R3 and let x ,x ,x be Definition 5.1. (1) The difference of matrix is a + 1 2 3 1 2 3 " c d # coordinates in R3. d − b − c. We know that a SNI matrix S yields a centrally symmetric alcoved (2) The 2–minors of a matrix A = [aij ] are the differences of polyhedron P(S), by lemma 3.3. the order 2 submatrices of A. If i,j,k,l ∈ [n], i < j and k < l, we write Lemma 7.1 (Polar of a centrally symmetric alcoved poly- ∈ SNI hedron). If S = [sij ] M4 with sij < 0, all i , j, then aij;kl := aik + ajl − ail − ajk , and (9) ! ◦ vi vi − vj (P(S)) = conv ± ,± : i,j ∈ [3],i , j . (13) aij;kl = −aji;kl = −aij;lk = aji;lk . (10) si4 sij If i = j or k = l, we write aij;kl = 0. Proof. We know that sij = sji ≤ 0. The alcoved polyhedron Straightforward computations yield the following. P(S) is defined by si4 ≤ xi ≤ −si4 and sij ≤ xi − xj ≤ −sij , or equivalently, 0 −2 − e12 −2 − e13 −2 * + * + −2 − e21 0 −2 − e23 −2 x v xi − xj vi − vj V =   (16) − ≤ i i ≤ − ≤ ≤  −2 − e31 −2 − e32 0 −2  1 = x, 1 and 1 = x, 1,   si4 si4 sij sij 0 0 0 0   T   i,j ∈ [3], i , j, whence the result follows.  with E = E . For simplicity of notation, we write x = e23, y = e13   and z = e12, with A normal matrix without zeros outside the diagonal (as in the − 1 ≤ z ≤ y ≤ x ≤ 0. (17) former lemma) is called strictly normal. The cant sequence isc = (x,y,z,x,y,z) andm = (|x |, |y|, |x|, |x|, |y|, |x |), M = (|y|, |z|, |z|, |y|, |z|, |z|). Using formula (12), we get 1     vol(P) = 8 − x2 (y + z) − y2z + 2x3 + y3 − 2 x2 + y2 + z2 . 3 (18) Assume y = x = 0. We get 2 vol(P|y=x=0) = 8 − 2z .

Indeed, P|y=x=0 is the unit cube Q (of volume 8) canted by the two planes of equations x − x 2 1 = ±1. 2 + z Since the polar of Q is an octahedron, then the polar of P|y=x=0 is ◦ Figure 7: Unit square centered at O and its polar. The origin Q with two additional vertices: v − v is marked in red ± 2 1 . 2 + z Since both P and P◦ are symmetric, we look at the upper half of x2−x1 these bodies. There, the plane 2+z = 1 yields a 4–gonal facet in 8 MAHLER CONJECTURE HOLDS FOR v2−v1 ◦ P, whence the vertex 2+z yields 4 concurrent facets in P . The ALCOVED POLYHEDRA difference between P◦ and Q◦ is that tetrahedra with vertices 0, ◦ v2−v1 v2−v1 The Mahler volume product of P is the product vol(P) vol(P ), by 2+z , v2, ±v3 and tetrahedra with vertices 0, 2+z , −v1, ±v3 have ◦ definition. It is well known that Mahler volume product is invariant been added (to Q ) and tetrahedra with vertices 0,v2, −v1, ±v3 have with respect to affine–linear transformations and homotheties. For been removed (from Q◦). The volume of any of the added tetrahedra 1 1 a centrally symmetric convex body, the Mahler conjecture is is 6(2+z) and the volume of any of the removed tetrahedra is 6 . So, ◦ ◦ 3 when passing from Q to P , we have a total volume gain of ◦ 4 ! vol(P) vol(P ) ≥ (14) 4 2 −2z 3! 2 − = ( + z) ( + z) with equality if and only if P is a box. It dates back to 1938. A 6 2 6 3 2 recent survey on the conjecture is [16]; see also [11, 24] and the and we get bibliography therein. A proof of the conjecture in 3–dimensional ◦ 4 ◦ 4 −2z vol((Q) ) = , vol((P|y=x= ) ) = + . space is announced in [12]. 3 0 3 3(2 + z) In this section we show that Mahler conjecture holds true, for Assume now x = 0. From (18), we have centrally symmetric alcoved polyhedra. 3 2 2 y vol(P|x=0) = 8 − 2z − (2 + z)y + . First, we compute the volume of the polar of a centrally symmet- 3 P ∈ SNI Indeed, P|x=0 is the unit cube Q canted by the 4 planes of equations ric alcoved polyhedron . We write a defining matrix S M4 for P and, if V = DS is a conjugate of S, then vol P = vol P(V ). x2 − x1 x3 − x1 = ±1, = ±1. (19) Now corollary 6.2 applies to V , because the symmetry is transferred 2 + z 2 + y from S to E = B − V (by item 1 in lemma 6.1). We look at the upper half of P and P◦ and note that two vertices It is no restriction to assume that P is maximal with respect appear in the polar P◦ SNI to f –vector. Then the matrix S ∈ M satisfies sij , 0, unless 4 v2 − v1 v3 − v1 i = j. By affine invariance of the Mahler volume product, wemay , . (20) 2 + z 2 + y assume that the bounding box of P is the unit cube (of edge–length The planes (19) yield a 4–gon and an adjacent 5–gon as facets in P. 2), centered at the origin. Further, we may assume that −1 ≤ e12 ≤ Thus, the vertices (20) yield a 4–pyramid and an adjacent 5–pyramid e13 ≤ e23 ≤ 0, without loss of generality. Thus, our matrices are in P◦. Some computations show that 0 −2 − e12 −2 − e13 −1 ◦ 4 −3y − 4z − 3yz −2 − e 0 −2 − e −1 vol((P|x=0) ) = + д(y,z), д(y,z) := . S =  21 23  (15) 3 3(2 + y)(2 + z)  −2 − e31 −2 − e31 0 −1    −  −1 −1 −1 0  Note that д(0,z) = 2z agrees with the case y = x = 0 above.   3(2+z)     ◦ 4 General case: similar computations show that vol(P ) = 3 + we add from 1 to 6, getting the following expression h(x,y,z) h(x,y,z) = 2/3 + 2/3 + 2/3 + with : (2+x )(2+y) (2+y)(2+z) (2+z)(2+x ) 5 5 5 4 2 4 MC = 192w w2 + 336w w3 + 432w w4 + 768w w + 2112w w2w3 1/3 2/3 − 1 1 1 1 2 1 2+y + 2+z 1 and h(0,y,z) = д(y,z). We have proved 4 4 2 4 4 2 +2472w1w2w4 + 1152w1w3 + 2568w1w3w4 + 1224w1w4 +1200w3w3 + 4524w3w2w + 5076w3w2w Theorem 8.1. If P ⊂ R3 is the centrally symmetric alcoved poly- 1 2 1 2 3 1 2 4 3 2 3 3 2 hedron given by the inequalities −1 ≤ xj ≤ 1, j ∈ [3], −2 − z ≤ +4824w1w2w3 + 10440w1w2w3w4 + 4992w1w2w4 x − x ≤ 2 + z, −2 − x ≤ x − x ≤ 2 + x, −2 −y ≤ x − x ≤ 2 +y, 3 3 3 2 3 2 3 3 1 2 2 3 3 1 +1528w1w3 + 4896w1w3w4 + 4740w1w3w4 + 1380w1w4 then 2 4 2 3 2 3 2 2 2 +912w1w2 + 4392w1w2w3 + 4830w1w2w4 + 6960w1w2w3 2 2 2 2 2 2 3 +14850w1w2w3w4 + 7146w1w2w4 + 4442w1w2w3 ◦ 1 2/3 2/3 2/3 1/3 2/3 vol(P ) = + + + + + . 2 2 2 2 2 3 3 (2 + x)(2 + y) (2 + y)(2 + z) (2 + z)(2 + x) 2 + y 2 + z +14034w1w2w3w4 + 13656w1w2w3w4 + 4050w1w2w4 (21) 2 4 2 3 2 2 2 2 3 +972w1w3 + 4092w1w3w4 + 6072w1w3w4 + 3702w1w3w4 2 4 5 4 4 +774w1w4 + 336w1w2 + 1980w1w2w3 + 2160w1w2w4 3 2 3 3 2 +4176w1w2w3 + 8862w1w2w3w4 + 4302w1w2w4 0 −2 − z −2 − y −1 + w w2w3 + w w2w2w + w w2w w2 −2 − z 0 −2 − x −1 4030 1 2 3 12657 1 2 3 4 12414 1 2 3 4 Proof. P = P(S) with S =   . 2 3 4 3 2 2  −2 − y −2 − x 0 −1  +3744w1w w + 1790w1w2w + 7475w1w2w w4 + 11163w1w2w w   2 4 3 3 3 4  −1 −1 −1 0  +6918w w w w3 + 1482w w w4 + 292w w5 + 1534w w4w   1 2 3 4 1 2 4 1 3 1 3 4   3 2 2 3 4 5   +3120w1w3w4 + 2988w1w3w4 + 1326w1w3w4 + 216w1w4 6 5 5 4 2 4 +48w2 + 336w2w3 + 366w2w4 + 888w2w3 + 1884w2w3w4 4 2 3 3 3 2 3 2 Using (18) and (21), and clearing denominators, we transform +924w2w4 + 1152w2w3 + 3615w2w3w4 + 3582w2w3w4 Mahler conjecture (14) into the question of whether the below de- 3 3 2 4 2 3 2 2 2 +1098w2w4 + 776w2w3 + 3233w2w3w4 + 4875w2w3w4 fined polynomial MC is non–negative on the simplex S given by 2 3 2 4 5 4 P6 +3072w w3w + 672w w + 256w2w + 1340w2w w4 −1 ≤ z ≤ y ≤ x ≤ 0, where MC = MCj , with MCj homoge- 2 4 2 4 3 3 j=1 3 2 2 3 4 5 neous in x,y,z of degree j: +2752w2w3w4 + 2682w2w3w4 + 1218w2w3w4 + 204w2w4 6 5 4 2 3 3 2 4 +32w3 + 204w3w4 + 540w3w4 + 728w3w4 + 516w3w4 4 3 2 3 2 4 3 2 5 6 MC6 = 2x yz − 3x y z − 3x yz + xy z − 3xy z , +180w3w4 + 24w4 . MC = x4y + x4z − x3y2 − x3yz − x3z2 5 8 6 12 23 9 This is a certificate of non–negativeness of MC on S, since all the 2 2 2 2 −6x y z − 6x yz coefficients are non–negative. The only missing term in MC (as a 4 3 2 2 3 4 3 2 6 +4xy − 15xy z − 9xy z − 6xyz + 2y z − 6y z , homogeneous polynomial in w1,w2,w3,w4) is w1. This means that the only real root of MC is given by w = 1 and w = w = w = 0, MC = 24x4 − 40x3y − 38x3z − 30x2y2 − 66x2yz 1 2 3 4 4 equivalently, by x = y = z = 0. This shows that equality is only 2 2 3 2 −24x z − 12xy − 54xy z attained by boxes, among centrally–symmetric alcoved polyhedra. −24xyz2 − 18xz3 + 10y4 − 34y3z − 24y2z2 − 12yz3, We have proved Mahler conjecture for alcoved polyhedra. The con- 3 2 2 2 jecture also holds for limits of centrally symmetric alcoved polyhe- MC3 = −8x − 156x y − 144x z − 72xy − 72xyz dra.  −72xz2 − 28y3 − 144y2z − 60yz2 − 48z3, Inspiration came from [9] and 2.24 in [10], for the former proof. 2 2 2 MC2 = −192x − 96xy − 120xz − 192y − 144yz − 192z , MC1 = −96x − 144y − 192z. 9 SUMMARY, FINAL REMARKS AND FUTURE DEVELOPMENTS

In order to answer the question MC|S ≥ 0, we consider the poly- We have computed the volume of alcoved polyhedra and we have nomials w1 = 1 + z, w2 = y − z, w3 = x − y, w4 = −x, which verified Mahler conjecture. Geometry tells us that our polyhedron determine the simplex S and are non–negative on S. We have is a perturbed box, more precisely, a canted box. In general, it is x = −w4, y = −w3 − w4, z = −w2 − w3 − w4 and the relation a dodecahedron with 20 vertices and 30 edges. Our method is to represent a d–dimensional polytope by a normal idempotent square matrix of order n = d + 1. Then, geometric questions about the 1 = w1 + w2 + w3 + w4. (22) polytope are answered through matrix computations. We have worked out the case d = 3. By a translation of the polyhedron, the vertex having larger coordinates can be moved to the origin. The ∈ 6−j For each j [6], we compute MCj = 1 MCj = (w1 + w2 + w3 + corresponding matrix is then visualized, in which case the matrix 6−j ′ w4) MCj , a degree 6 polynomial, homogeneous in the wjs, and splits as a (classical) sum of a box matrix and a perturbation matrix. The entries of these two matrices are precisely the edge–lengths of REFERENCES the polyhedron. The volume formula follows from here. [1] M. Akian, R. Bapat, and S. Gaubert. 2007. Max–plus algebra. In Handbook of What is specific of our method in dimension 3 is a natural ques- linear algebra, L. Hobgen (Ed.). Chapman and Hall, Chapter 25. [2] F. Ardila, C. Benedetti, and J. Doker. 2010. Matroid polytopes and their volumes. tion to ask. The particularities of dimensions d = 2 and 3 are various. Discrete and Computational Geometry 43, 4 (June 2010), 841–854. arXiv:0810. First, we know the f–vectors of maximal alcoved polytopes, which 3947v3(2011) , = , , = [3] A. Ben-Israel. 1992. A volume associated with m × n matrices. 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