LOWER BOUNDS ON THE COEFFICIENTS OF EHRHART POLYNOMIALS MARTIN HENK AND MAKOTO TAGAMI Abstract. We present lower bouns for the coefficients of Ehrhart polynomials of convex lattice polytopes in terms of their volume. We also introuce two formulas for calculating the Ehrhart series of a kin of a weak free sum of two lattice polytopes an of integral ilates of a polytope. As an application of these formulas we show that Hibi s lower boun on the coefficients of the Ehrhart series is not true for lattice polytopes without interior lattice points.. Introuction Let P be the set of all convex -imensional lattice polytopes in the - imensional Eucliean space R with respect to the stanar lattice Z, i.e., all vertices of P P have integral coorinates an im(p =. The lattice point enumerator of a set S R, enote by G(S, counts the number of lattice (integral points in S, i.e., G(S = #(S Z. In 962, Eugéne Ehrhart (see e.g. [3, Chapter 3], [7] showe that for k N the lattice point enumerator G(k P, P P, is a polynomial of egree in k where the coefficients g i (P, 0 i, epen only on P : (. G(k P = g i (P k i. The polynomial on the right han sie is calle the Ehrhart polynomial, an regare as a formal polynomial in a complex variable z C it is enote by G P (z. Two of the + coefficients g i (P are almost obvious, namely, g 0 (P =, the Euler characteristic of P, an g (P = vol(p, where vol( enotes the volume, i.e., the -imensional Lebesgue measure on R. It was shown by Ehrhart (see e.g. [3, Theorem 5.6], [8] that also the secon leaing coefficient amits a simple geometric interpretation as lattice surface area of P (.2 g (P = vol (F 2 et(afff Z. F facet of P Here vol ( enotes the ( -imensional volume an et(afff Z enotes the eterminant of the ( -imensional sublattice containe in the affine hull of F. All other coefficients g i (P, i 2, have no such known 2000 Mathematics Subject Classification. 52C07, H06. Key wors an phrases. Lattice polytopes, Ehrhart polynomial. The secon author was supporte by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
2 MARTIN HENK AND MAKOTO TAGAMI geometric meaning, except for special classes of polytopes. For this an as a general reference on the theory of lattice polytopes we refer to the recent book of Matthias Beck an Sinai Robins [3] an the references within. For more information regaring lattices an the role of the lattice point enumerator in Convexity see [9]. In [4, Theorem 6] Ulrich Betke an Peter McMullen prove the following upper bouns on the coefficients g i (P in terms of the volume: g i (P ( i i stirl(, i + stirl(, ivol(p + (, i =,...,, (! where stirl(, i enote the Stirling numbers of the first kin. In orer to present our lower bouns on g i (P in terms of the volume we nee some notation. For an integer i an a variable z we consier the polynomial ( z + i (z + i(z + i... (z + i ( =!, i an we enote its r-th coefficient by Cr,i, 0 r. For instance, it is C,i =, an for 0 i we have C0,i = 0. For 3 we are intereste in (.3 M r, = min{c r,i : i 2}. Obviously, we have M, = an it is also easy to see that (.4 M, = C, 3 = (. 2 With the help of these numbers M r, we obtain the following lower bouns Theorem.. Let P P, 3. Then for i =,..., we have g i (P } {( i stirl( +, i + + (!vol(p M i,.! In the case i =, for instance, we get together with (.4 the boun { g (P 3 } (! 2!vol(P. Since the lattice surface area of any facet is at least /(! we have the trivial inequality (cf. (.2 (.5 g (P + 2 (!. Hence the lower boun on g (P is only best possible if vol(p = /!. In the cases i {, 2, 2}, however, Theorem. gives best possible bouns for any volume
LOWER BOUNDS ON THE COEFFICIENTS OF EHRHART POLYNOMIALS 3 Corollary.2. Let P P. Then i g (P + 2 + + 2 + 2 ( 2!vol(P, ii iii g 2 (P ( {stirl( +, 3 + (( 2! + stirl(, 2 (!vol(p },! ( (+! 24 {3( +!vol(p } : o, g 2 (P (! 24 {3( + 2 ( 2!vol(P } : even. An the bouns are best possible for any volume. For some recent inequalities involving more coefficients of Ehrhart polynomials we refer to [2]. Next we come to another family of coefficients of a polynomial associate to lattice polytopes. The generating function of the lattice point enumerator, i.e., the formal power series Ehr P (z = k 0 G P (k z k, is calle the Ehrhart series of P. It is well known that it can be expresse as a rational function of the form Ehr P (z = a 0(P + a (P z + + a (P z ( z +. The polynomial in the numerator is calle the h -polynomial. Its egree is also calle the egree of the polytope [] an it is enote by eg(p. Concerning the coefficients a i (P it is known that they are integral an that a 0 (P =, a (P = G(P ( +, a (P = G(int(P, where int( enotes the interior. Moreover, ue to Stanley s famous nonnegativity theorem (see e.g. [3, Theorem 3.2], [6] we also know that a i (P is non-negative, i.e., for these coefficients we have the lower bouns a i (P 0. In the case G(int(P > 0, i.e., eg(p =, these bouns were improve by Takayuki Hibi [2] to (.6 a i (P a (P, i eg(p. In this context it was a quite natural question whether the assumption eg(p = can be weaken (see e.g. [4], i.e., whether these lower bouns (.6 are also vali for polytopes of egree less than. As we show in Example. the answer is alreay negative for polytopes having egree 3. The problem in orer to stuy such a question is that only very few geometric constructions of polytopes are known for which we can explicitly calculate the Ehrhart series. In [3, Theorem 2.4, Theorem 2.6] the Ehrhart series of special pyramis an ouble pyramis over a basis Q are etermine in terms of the Ehrhart series of Q. In a recent paper Braun [6] gave a very nice prouct formula for the Ehrhart series of the free sum of two lattice polytopes, where one of the polytopes has to be reflexive. Here we consier the following construction, which might be regare as a very weak or fake free sum.
4 MARTIN HENK AND MAKOTO TAGAMI Lemma.3. For P P p an Q P q let P Q = conv {(x, 0 q, 0, (0 p, y, : x P, y Q} P p+q+, where 0 p an 0 q enote the p- an q-imensional 0-vector, respectively. Then Ehr P Q (z = Ehr P (z Ehr Q (z. In orer to apply this Lemma we consier two families of lattice simplices. For an integer m N let T (m = conv{o, e, e + e 2, e 2 + e 3,..., e 2 + e, e + m e }, S (m = conv{o, e, e 2, e 3,..., e, m e }, where e i enotes the i-th unit vector. It was shown in [4] that (.7 Ehr (m T (z = + (m z 2 ( z + an Ehr (m S (z = Example.. For q N o an l, m N we have Ehr T (l+ q S (m+ p (z = + m z + l z q+ 2 + m l z q+3 2 ( z p+q+2. + (m z ( z +. In particular, for q 3 an l < m this shows that (.6 is, in general, false for lattice polytopes without interior lattice points. Another formula for calculating the Ehrhart Series from a given one concerns ilates. Here we have Lemma.4. Let P P, k N an let ζ be a primitive k-th root of unity. Then Ehr k P (z = k Ehr P (ζ i z k. k The lemma can be use, for instance, to calculate the Ehrhart series of the cube C = {x R : x i, i }. Example.2. For two integers j,, 0 j, let A(, j be the Eulerian numbers (see e.g. [3, pp. 28] an for convenience we set A(, j = 0 if j / {0,..., }. Then + ( + a i (C = A(, 2 i + j, 0 i. j j=0 Of course, the cube C may be also regare as a prism over a ( cube, an as a counterpart to the bipyrami construction in [3] we calculate here also the Ehrhart series of some special prism.
LOWER BOUNDS ON THE COEFFICIENTS OF EHRHART POLYNOMIALS 5 Example.3. Let Q P, m N, an let P = {(x, m : x Q} be the prism of height m over Q. Then a i (P = (m i + a i (Q + (m( i + a i (Q, 0 i, where we set a (Q = a (Q = 0. It seems to be quite likely that for the class of 0-symmetric lattice polytopes Po the lower bouns on a i (P can consierably be improve. In [5] it was conjecture that for P Po ( a i (P + a n i (P (a n (P +, i where equality hols for instance for the cross-polytopes C (2 l = conv{±l e, ±e i : 2 i }, l N, with 2l interior lattice points. It is also conjecture that these cross-polytopes have minimal volume among all 0-symmetric lattice polytopes with a given number of interior lattice points. The maximal volume of those polytopes are known by the work of Blichfelt an van er Corput (cf. [9, p. 5] an, for instance, the maximum is attaine by the boxes Q (2 l = { x l, x i, 2 i } with 2 l interior points. By the Examples.2 an.3 we can easily calculate the Ehrhart series of these boxes Example.4. Let l N. Then, for 0 i, a i (Q (2 l = (2 l i + a i (C + (2 l( i + a i (C. It is quite tempting to conjecture that these numbers form the corresponing upper bouns on a i (P + a n i (P for 0-symmetric polytope with 2 l interior lattice points. In the 2-imensional case this follows easily from a result of Paul Scott [5] which implies that a (P 6 l = a (Q 2 (2 l for any 0-symmetric convex lattice polygon with 2 l interior lattice points. Concerning lower bouns on g i (P for 0-symmetric polytopes P we only know, except the trivial case i =, a lower boun on g (P (cf. (.5. Namely g (P g (C = 2 (!, where C = conv{±e i : i } enotes the regular cross-polytope. This follows immeiately from a result of Richar P. Stanley [7, Theorem 3.] on the h-vector of symmetric Cohen-Macaulay simplicial complex. Motivate by a problem in [] we stuy in the last section also the relate question to boun the surface area F(P of a lattice polytope P. To this en let T = conv{0, e,..., e } be the stanar simplex. Proposition.5. Let P P, im P =. Then F(P F(C = 2! 3 2 : P = P, F(T = + (! : otherwise.
6 MARTIN HENK AND MAKOTO TAGAMI The paper is organize as follows. In the next section we give the proof of our main Theorem.. Then, in Section 3, we prove the Lemmas.3 an.4 an show how the Ehrhart series in the Examples. an.2 can be euce. Moreover, we show that some recent bouns of Jaron Treutlein [8] on the coefficients of h -polynomials of egree 2 give inee a complete classification of all these h -polynomials (cf. Proposition 3.2. Finally, in the last section we provie a proof of Proposition.5 which in the symmetric cases is base on a isoperimetric inequality for cross-polytopes (cf. Lemma 4.. 2. Lower bouns on g i (P In the following we enote for an integer r an a polynomial f(x the r-th coefficient of f(x, i.e. the coefficient of x r, by f(x r. Before proving Theorem. we nee some basic properties of the numbers C r,i an M r, efine in the introuction (see (.3. Lemma 2.. i Cr,i = ( r Cr, i for 0 i. ii Let 3. Then M r, 0 for r <. Proof. For i we just note that Cr,l is the ( r-th elementary symmetric function of {l, l,..., l ( }. On account of i it suffices to prove ii when r is even an we o that by inuction on. For = 3 an r = we have M,3 = C, 3 =. So let > 3, an since = 0 we may also assume r. It is easy to see that C 0,i (2. C r,i = (i + C r,i + C r,i, an by inuction we may assume that there exists a j {,..., 3} with Cr,j 0. Observe that (r is even. If C r,j 0 we obtain by (2. that Cr,j 0 an we are one. So let C r,j < 0. By part i we know that C r,j = ( r C r, 2 j an C r,j = ( r C r, 2 j. Since r is even we conclue C r, 2 j > 0 an C r, 2 j account of (2. we get Cr, 2 j 0 an so M r, 0. 0. Hence, on Proof of Theorem.. We follow the approach of Betke an McMullen use in [4, Theorem 6]. By expaning the Ehrhart series at z = 0 one gets (see e.g. [3, Lemma 3.4] ( z + i (2.2 G P (z = a i (P. In particular, we have (2.3! a i (P = g (P = vol(p.
LOWER BOUNDS ON THE COEFFICIENTS OF EHRHART POLYNOMIALS 7 For short, we will write a i instea of a i (P an g i instea of g i (P. With these notation we have ( z + i!g r =!G P (z r =! a i (2.4 r = Cr, + (a Cr, + a Cr,0 + a i Cr, i. Since Cr, 0 we get with Lemma 2. i that C r, = C r,0. Together with a a an Cr, = ( r stirl( +, r + we fin (2.5!g r ( r stirl( +, r + + a i Cr, i i=2 i=2 = ( r stirl( +, r + + a i (C r, i M r, + i=2 (a + a M r, ( r stirl( +, r + + (!vol(p M r,, a i M r, where the last inequality follows from the efinition of M r, an the negativity of M r, (cf. Lemma 2. ii. For r one can easily show that the numbers C r,, M r, are negative an so the proof above (cf. (2.4 an (2.5 gives!g r ( r stirl( +, r + + 2 a (P + (!vol(p M r, = ( r stirl( +, r + 2( + + 2G(P + (!vol(p M r,. In orer to verify the inequalities in Corollary.2 we have to calculate the numbers M r, for r =, 2, 2. Proposition 2.2. Let 3. Then i M, = C, 2 = ( 2!, ii M 2, = C2, 2 = ( 2! + ( stirl(, 2, C = ( +, iii M 2, = 4 3, o, 2 = 4( 3, even. C, 2 Proof. C,i is the -st elementary symmetric function of {i,..., 0,..., i ( }. Thus C,i = ( i i! ( i! an M, = min{c,i : i 2} = C, 2 = ( 2! In the case r = 2 we obtain by elementary calculations that C2,i = i!stirl( i, 2 + ( ( i!stirl(i +, 2, from which we conclue M 2, = C2, 2 = ( 2! + ( stirl(, 2.
8 MARTIN HENK AND MAKOTO TAGAMI For the value of M 2, we first observe that C 2,i C 2,i = (z + i (z + i... (z + i ( 2 = = (z + i... (z + i ( (z + i 2 i j= +i+ j (i ( + i = ( ( + 2 i. 2 i j= +i+ Thus the function C 2,i is ecreasing in 0 i /2 an increasing in /2 i. So it takes its minimum at i = /2. First let us assume that is o. Then ( z + ( /2 M 2, = C =! 2, 2 2 ( /2 = z (z 2 (z 2 4... (z 2 (( /2 2 = i 2 2 = 4 ( + 3. The even case can be treate similarly. Now we are able to prove Corollary.2 Proof of Corollary.2. The inequalities just follow by inserting the value of M r, given in Proposition 2.2 in the general inequality of Theorem.. Here we also have use the ientities stirl( +, 2 = ( +! i an stirl( +, = 3 + 2 4 j ( + It remains to show that the inequalities are best possible for any volume. For r = 2 we consier the simplex T (m (cf. (.7 with a 0 (T (m =, a /2 (T (m = (m an a i (T (m = 0 for i / {0, /2 }. Then vol(t (m = m/! an on account of Proposition 2.2 we have equality in (2.4 an (2.5. (m For r =, 2 an 4 we consier the ( 4-fol pyrami T over T (m 4 (m given by T = conv{t (m (m 4, e 5,..., e }. Then vol( T = m/! an in view of (.7 an [3, Theorem 2.4] we obtain a 0 ( (m T =, a 2 ( (m (m T = m an a i ( T = 0, i / {0, 2}. Again, by Proposition 2.2 we have equality in (2.4 an (2.5. In the 3- imensional case it remains to show that the boun on g (P is best possible. For the calle Reeve simplex T (m g (R (m 3 = 2 m 6 3, however, it is easy to check that whereas vol(r(m 3 = m/6. 3.
LOWER BOUNDS ON THE COEFFICIENTS OF EHRHART POLYNOMIALS 9 3. Ehrhart series of some special polytopes We start with the short proof of Lemma.3. Proof of Lemma.3. Since Ehr P (z Ehr Q (z = k 0 ( m+l=k G P (mg Q (l it suffices to prove that the Ehrhart polynomial G P Q (k of the lattice polytope P Q P p+q+ is given by G P Q (k = m+l=k G P (mg Q (l. This, however, follows immeiately from the efinition since k (P Q = {λ (x, o q, 0 + (k λ (o p, y, : x P, y Q, 0 λ k}. z k, Example. in the introuction shows an application of this construction. For example.2 we nee Lemma.4. Proof of Lemma.4. With w = z k we may write k k Ehr P (ζ i w = k k G P (m(ζ i w m = k G P (mw m ζ i m. k m 0 Since ζ is a k-th root of unity the sum k ζi m is equal to k if m is a multiple of k an otherwise it is 0. Thus we obtain m 0 k Ehr P (ζ i w = G P (m kw m k = G k P (mz m = Ehr k P (z. k m 0 m 0 As an application of Lemma.4 we calculate the Ehrhart series of the cube C (cf. Example.2. Instea of C we consier the translate cube 2 C, where C = {x R : 0 x i, i }. In [3, Theorem 2.] it was shown that a i ( C = A(, i + where A(, i enotes the Eulerian numbers. Setting
0 MARTIN HENK AND MAKOTO TAGAMI w = z Lemma.4 leas to Ehr C (z = ( Ehr 2 C (w + Ehr C ( w ( = A(, i wi A(, i ( wi+ 2 ( w + + ( + w + ( = 2 ( z + A(, i w i ( + w + + A(, i ( w i+ ( w + = ( z + A(, i Substituting 2 l = i + j gives ( Ehr C (z = ( z + l=0 = ( z + l=0 2 l+ i=2 l z l + + j=0, i + j even j=0 ( + j w i+j ( + A(, i w 2 l 2 l + i ( + A(, 2 l + j, j which explains the formula in Example.2. In orer to calculate in general the Ehrhart series of the prism P = {(x, m : x Q} where Q P, m N (cf. Example.3, we use the ifferential operator T efine by z z. Consiere as an operator on the ring of formal power series we have (cf. e.g. [3, p. 28] (3. f(k z k = f(t z k 0 for any polynomial f. Since G P (k = (m k + G Q (k we euce from (3. Ehr P (z = (m T + Ehr Q (z = mz z Ehr Q(z + Ehr Q (z. Thus Ehr P (z = m z i a i(qz i ( z + a i(q z i ( z + + a i(q z i ( z = (m i + a i(qz i ( z + m a i(qz i+ ( z + = ( z + ((m i + a i (Q + (m( i + a i (Q z i, which is the formula in Example.3. In a recent paper Jaron Treutlein [8] generalize a result of Scott [5] to all egree 2 polytopes by showing
LOWER BOUNDS ON THE COEFFICIENTS OF EHRHART POLYNOMIALS Theorem 3. (Treutlein. Let P P of egree 2 an let a i = a i (P. Then { 7, a 2 =, (3.2 a 3 a 2 + 3, a 2 2. The next proposition shows that these conitions inee classify all h polynomials of egree 2. Proposition 3.2. Let f(z = a 2 z 2 +a z +, a i N, satisfying the inequalities in (3.2. Then f is the h polynomial of a lattice polytope. Proof. We recall that a (P = G(P ( + an a (P = G(int(P for P P. In the case a 2 =, a = 7 the triangle conv{0, 3 e, 3 e 2 } has the esire h -polynomial. Next we istinguish two cases: i a 2 < a 3 a 2 + 3. For integers k, l, m with 0 l, k m + let P P 2 given by P = conv{0, l e, e 2 + (m + e, 2 e 2, 2 e 2 + k e }. Then it is easy to see that a 2 (P = m an P has n + l + 4 lattice points on the bounary. Thus a (P = n + l + m +. ii a a 2. For integers l, m with 0 l m let P P 3 given by P = conv{0, e, e 2, l e 3, e + e 2 + (m + e 3 }. The only lattice points containe in P are the vertices an the lattice points on the ege conv{0, l e 3 }. Thus a 3 (P = 0 an a (P = l. On the other han, since (l + m/6 = vol(p = ( 3 a i(p /6 (cf. (2.3 it is a 2 (P = m. 4. 0-symmetric lattice polytopes In orer to stuy the surface area of 0-symmetric polytopes we first prove an isoperimetric inequality for the class of cross-polytopes. Lemma 4.. Let v,..., v R be linearly inepenent an let C = conv{±v i : i }. Then F(C 2 vol(c! 3 2, an equality hols if an only if C is a regular cross-polytope, i.e., v,..., v form an orthogonal basis of equal length. Proof. Without loss of generality let vol(c = 2 /!. Then we have to show (4. F(C 2! 3 2. By stanar arguments from convexity (see e.g. [0, Theorem 6.3] the set of all 0-symmetric cross-polytopes with volume 2 /! contains a cross-polytope C = conv{±w,..., ±w }, say, of minimal surface area. Suppose that some of the vectors are not pairwise orthogonal, for instance, w an w 2. Then we apply to C a Steiner-Symmetrization (cf. e.g. [0, pp. 69] with respect to the hyperplane H = {x R : w i x = 0}. It is easy to check that the Steiner-symmetral of C is again a cross-polytope C, say, with vol( C = vol(c (cf. [0, Proposition 9.]. Since C was not symmetric with respect
2 MARTIN HENK AND MAKOTO TAGAMI to the hyperplane H we also know that F( C < F(C which contraicts the minimality of C (cf. [0, p. 7]. So we can assume that the vectors w i are pairwise orthogonal. Next suppose that w > w 2, where enotes the Eucliean norm. Then we apply Steiner-Symmetrization with respect to the hyperplane H which is orthogonal to w w 2 an bisecting the ege conv{w, w 2 }. As before we get a contraiction to the minimality of C. Thus we know that w i are pairwise orthogonal an of same length. By our assumption on the volume we get w i =, i, an it is easy to calculate that F(C = (2 /! 3/2. So we have F(C F(C = 2! 3 2, an by the foregoing argumentation via Steiner-Symmetrizations we also see that equality hols if an only C is a regular cross-polytope generate by vectors of unit-length. The etermination of the minimal surface area of 0-symmetric lattice polytope is an immeiate consequence of the lemma above, whereas the non-symmetric case oes not follow from the corresponing isoperimetric inequality for simplices. Proof of Proposition.5. Let P P with P = P an let im P =. Then P contains a 0-symmetric lattice cross-polytopes C = conv{±v i : i }, say, an by the monotonicity of the surface area an Lemma 4. we get ( 2 3 (4.2 F(P F(C 2 vol(c.! Since v i Z, i, we have vol(c = (2 /! et(v,..., v 2 /!, which shows by (4.2 the 0-symmetric case. In the non-symmetric case we know that P contains a lattice simplex T = {x R : a i x b i, i + }, say. Here we may assume that a i Z n are primitive, i.e., conv{0, a i } Z n = {0, a i }, an that b i Z. Furthermore, we enote the facet P {x R : a i x = b i } by F i, i +. With these notations we have et(afff i Z n = a i (cf. [3, Proposition.2.9]. Hence there exist integers k i with (4.3 vol (F i = k i a i (!, an so we may write + F(P F(T = vol (F i + (! a i. We also have + vol (F i a i / a i = 0 (cf. e.g. [0, Theorem 8.2] an in view of (4.3 we obtain + k i a i = 0. Thus, since the + lattice vectors a i
LOWER BOUNDS ON THE COEFFICIENTS OF EHRHART POLYNOMIALS 3 are affinely inepenent we get (4.4 + a i 2 2. Together with the restrictions a i, i +, it is easy to argue that + a i is minimize if an only if norms a i are equal to an one is equal to. For instance, the intersection of the cone {x R + : x i, i + } with the hyperplane H α = {x R + : + x i = α}, α +, is the -simplex T (α with vertices given by the permutations of the vector (,...,, α of length + (α 2. Therefore, a vertex of that simplex is containe in {x R + : + x2 i 2} if α +. In other wors, we always have + a i +, which gives the esire inequality in the non-symmetric case (cf. (4.3. We remark that the proof also shows that equality in Proposition.5 hols if an only if P is the o-symmetric cross-polytope C or the simplex T (up to lattice translations. Acknowlegement. The authors woul like to thank Christian Haase for valuable comments. References [] V. V. Batyrev, Lattice polytopes with a given h -polynomial, Contemporary Mathematics, no. 423, AMS, 2007, pp. 0. [2] M. Beck, J. De Loera, M. Develin, J. Pfeifle, an R.P. Stanley, Coefficients an roots of Ehrhart polynomials, Contemp. Math. 374 (2005, 5 36. [3] M. Beck an S. Robins, Computing the continuous iscretely: Integer-point enumeration in polyhera, Springer, 2007. [4] U. Betke an P. McMullen, Lattice points in lattice polytopes, Monatsh. Math. 99 (985, no. 4, 253 265. [5] Ch. Bey, M. Henk, an J.M. Wills, Notes on the roots of Ehrhart polynomials, Discrete Comput. Geom. 38 (2007, 8 98. [6] B. Braun, An Ehrhart series formula for reflexive polytopes, Electron. J. Combinatorics 3 (2006, no., Note 5. [7] E. Ehrhart, Sur les polyères rationnels homothétiques à n imensions, C. R. Aca. Sci., Paris, Sér. A 254 (962, 66 68. [8], Sur un problème e géométrie iophantienne linéaire, J. Reine Angew. Math. 227 (967, 25 49. [9] P. M. Gruber an C. G. Lekkerkerker, Geometry of numbers, secon e., vol. 37, North- Hollan Publishing Co., Amsteram, 987. [0] P.M. Gruber, Convex an iscrete geometry, Grunlehren er mathematischen Wissenschaften, vol. 336, Springer-Verlag Berlin Heielberg, 2007. [] M. Henk an J.M. Wills, A Blichfelt-type inequality for the surface area, to appear in Mh. Math. (2007, Preprint available at http://arxiv.org/abs/0705.2088. [2] T. Hibi, A lower boun theorem for Ehrhart polynomials of convex polytopes, Av. Math. 05 (994, no. 2, 62 65.
4 MARTIN HENK AND MAKOTO TAGAMI [3] J. Martinet, Perfect lattices in Eucliean spaces, Grunlehren er Mathematischen Wissenschaften [Funamental Principles of Mathematical Sciences], vol. 327, Springer-Verlag, Berlin, 2003. [4] B. Nill, Lattice polytopes having h -polynomials with given egree an linear coefficient, (2007, Preprint available at http://arxiv.org/abs/0705.082. [5] P. R. Scott, On convex lattice polygons, Bull. Austral. Math. Soc. 5 (976, no. 3, 395 399. [6] R.P. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math. 6 (980, 333 342. [7], On the number of faces of centrally-symmetric simplicial polytopes, Graphs an Combinatorics 3 (987, 55 66. [8] J. Treutlein, Lattice polytopes of egree 2, (2007, Preprint availabel at http://arxiv. org/abs/0706.478. Martin Henk, Universität Mageburg, Institut für Algebra un Geometrie, Universitätsplatz 2, D-3906 Mageburg, Germany E-mail aress: henk@math.uni-mageburg.e Makoto Tagami, Universität Mageburg, Institut für Algebra un Geometrie, Universitätsplatz 2, D-3906 Mageburg, Germany E-mail aress: tagami@kenroku.kanazawa-u.ac.jp