Tropical Ehrhart theory and tropical volume

We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition is complemented by a brief study of arising complexity questions.


Introduction
Tropical geometry is the study of piecewise-linear objects defined over the (max, +)semiring that arises by replacing the classical addition '+' with 'max' and multiplication '·' with '+.' While this often focuses on combinatorial properties, see [11,25], we are mainly interested in metric properties. Measuring quantities from tropical geometry turned out to be fruitful for a better understanding of interior point methods for linear programming [2] and principal component analysis of biological data [37]. Moreover, it has interesting connections with representation theory [29,38] and computational complexity [22].
Driven by this motivation, we develop a new definition of a volume for tropical convex sets by a thorough investigation of the tropical analog of lattice point counting. This continues the investigation of intrinsic tropical metric properties that started around a tropical isodiametric inequality [15] and tropical Voronoi diagrams [14].
Tropical polytopes are finitely generated tropical convex sets, see (2) in Sect. 2.1. Former work only considered the lattice points Z d in a d-dimensional polytope, see in particular [12]. This idea was used to measure its Euclidean volume and deduce the hardness to compute it by counting the integer lattice points [22]. These lattice points arise naturally through the representation of affine buildings as tropical polytopes [29]. However, we are more interested in lattice points which are conformal with the semiring structure. Varying the semiring as explained in Sect. 2.3 leads to two natural notions: integer lattice points in polytopes over the (max, ·)-semiring and their image under a logarithm map over the (max, +)-semiring. This is related to the concept arising from 'dequantization,' but we show in Sect. 4.3 how our tropical volume concept differs from the existing ones [15].
The main idea leading to our novel concept of tropical volume is the following: For a classical polytope P ⊆ R d , the Euclidean volume describes the asymptotic behavior of its Ehrhart function L(P, k) = # kP ∩ Z d , that is, the function that counts lattice points from Z d that are contained in the kth dilate of the polytope P. This discretization further refines if P is a lattice polytope, meaning that all its vertices belong to Z d . In fact, Ehrhart proved that in this case L(P, k) agrees with a polynomial of degree at most d, for every positive integral dilation factor k ∈ Z >0 (see [6,Ch. 3]): The polynomial on the right-hand side is known as the Ehrhart polynomial of P, and the crucial point for us is that vol(P) = lim k→∞ L(P, k) k d = c d (P). Now, our approach toward an intrinsic tropical volume concept is to turn this discretization process around and to establish tropical analogs to the previously described classical ideas. This will be done in four steps: (i) We define a suitable concept of tropical lattice (depending on a fineness parameter) and tropical lattice polytopes in Sect. 2.2. (ii) In Sect. 3, we develop a tropical Ehrhart theory showing that the corresponding tropical Ehrhart function exhibits polynomial behavior. (iii) We then take the leading coefficient of the tropical Ehrhart polynomial as the definition of tropical volume. (iv) Finally, we extract the metric information that is independent of the fineness parameter of the tropical lattice by using its asymptotics and extend it to all tropical polytopes, without any integrality restriction. This is implemented in Sect. 4.
The development of our tropical Ehrhart theory rests on making the transition from the ring (R, +, ·) to the tropical semiring T = (R ∪ {−∞}, max, +) in two steps. More precisely, we first replace addition '+' by the maximum operation to obtain the semiring S (max,·) = (R ≥0 , max, ·). Then, for any b ∈ N ≥2 , the map x → log b (x) induces a semiring isomorphism between S (max,·) and T.
On the one hand, this point of view motivates us to introduce tropical integers as log b (Z ≥0 ), leading to what we call the tropical b-lattice log b (Z ≥0 ) d with fineness parameter b ∈ N ≥2 . And on the other hand, it allows to transfer classical Ehrhart theory on complexes of lattice polytopes to an Ehrhart theory for lattice polytopes over the various semirings which we explicitly describe in Theorems 3.2, 3.4, and 3.6. These results heavily rely on the interplay of the involved semirings associated with tropical geometry, cf. [11]. While this approach is very conceptual and offers a first understanding of tropical Ehrhart theory, it has the disadvantage of lacking a useful description of the coefficients of the resulting tropical Ehrhart polynomials.
Therefore, we take a second route based on the covector decomposition that allows to triangulate a tropical lattice polytope into so-called alcoved simplices which are both tropically and classically convex polytopes. This leads to the explicit representations of tropical Ehrhart coefficients in Theorem 3.14 and eventually to our desired intrinsic volume concept. The key insight here is that counting tropical lattice points in tropical dilations of alcoved simplices amounts to counting usual lattice points in dilates of diagonally transformed alcoved simplices (Lemma 3.11). To assemble the Ehrhart coefficients correctly from these pieces, we need a better understanding of lower-dimensional structures of the covector decomposition, which is achieved in Sect. 2.1.
As the result of the four-step-process outlined above, we define the tropical barycentric volume tbvol(P) of a tropical polytope P ⊆ T d as tbvol(P) := max where the maximum is taken over all points x ∈ P that are contained in a d-dimensional cell of the polyhedral complex associated to P. Our choice of name will become clear later on.
In Sect. 4.2, we investigate basic properties of the tropical barycentric volume. We prove that it satisfies the natural tropical analogs of the fundamental properties of the Euclidean volume: monotonicity, the valuation property, rotation invariance, homogeneity, non-singularity, and multiplicativity. In this sense, tbvol(·) is a meaningful and intrinsic volumetric concept for tropical geometry.
Furthermore, in Sect. 4.3 we compare the tropical barycentric volume with existing volumetric measures. For instance, it turns out to be bounded by the tropical dequantized volume qtvol + (·) defined in [15]. More precisely, if P = tconv(M) is the tropical polytope defined as the tropical convex hull of the columns of M ∈ T d×m , then we prove in Theorem 4.15 that tbvol(P) ≤ qtvol + (M). (1) Motivated by this inequality, we go a step further and work toward lower-dimensional volumetric measures in Sect. 5. We propose natural generalizations of the tropical barycentric volume that may serve as adequate tropical versions of the classical intrinsic volumes (or quermassintegrals) (cf. [35]). For example, we define a tropical lower barycentric i-volume tbvol − i (P) of P = tconv(M) and prove that it is upper bounded by the maximal tropical determinant of an (i × i)-submatrix of M (see Theorem 5.12). This extends (1), because qtvol + (M) can be defined as the maximal tropical determinant of a (d × d)-submatrix of M; see [15].
We close the paper with Sect. 6 in which we discuss computational aspects of the problem of computing the tropical barycentric volume. We argue that the decision problem that asks whether the tropical barycentric volume of a given tropical polytope is nonvanishing is equivalent to checking feasibility of a tropical linear program or to deciding winning positions in mean-payoff games. Therefore, this decision problem lies in NP ∩ coNP (cf. [22]). This equivalence is analogous to the classical setting, where existence of interior points in a polytope is equivalent to solving linear programs (cf. [24]). Based on the computation of the tropical barycenter of a tropical simplex, we moreover devise an algorithm to determine the tropical barycentric volume of a tropical d-polytope with m vertices, that runs in time O( m d+1 d 3 ).

Tropical convexity and tropical lattices
In this section, we fix the main notation of the paper, discuss the crucial concept of the itrunk of a tropical polytope, introduce the notion of tropical lattice leading to our tropical Fig. 1 The covector decomposition of a tropical 2-polytope. It consists of three quadrilaterals and a line segment, and it has nine pseudovertices. Its 2-trunk is obtained by cutting off the line segment connecting (1,1) to (1,0) Ehrhart theory, and finally review the relationship between different versions of convexity relevant to our studies.

Tropical polytopes and alcoved triangulations
We denote by T = (R ∪ {−∞}, ⊕, ) the max-tropical semiring, where ⊕ denotes the max operation and denotes the classical addition '+.' The tropical convex hull of a set V ⊆ T d is defined by If V is finite, this is called a tropical polytope. We will switch freely between matrices and the set of their columns. A set is tropically convex if it contains the tropical convex hull of each of its finite subsets. By the tropical Minkowski-Weyl theorem [21], there is a unique minimal set of points generating a tropical polytope; we call these points the vertices. The 'type decomposition' due to Develin and Sturmfels [17] shows that each tropical polytope has a decomposition into polytropes, which are classically and tropically convex polytopes [27]. Following [20], we use the name covector decomposition for this polyhedral complex formed by the polytropes. The vertices of the covector decomposition are called pseudovertices and the dimension of the tropical polytope is the maximal dimension of a polytope in the complex. Figure 1 depicts the covector decomposition of a tropical 2-polytope with vertices {(1, 0) , (2, 2) , (−2, 3) , (−2, 5) }.
For a tropical polytope P ⊆ T d , let the family of relatively open polytopes in the covector decomposition of P be denoted by F P . An element T ∈ F P is called an i-tentacle element, if it is not contained in the closure of any (i + 1)-dimensional polytope Q ∈ F P . In particular, the dimension of an i-tentacle element is smaller than or equal to i. The following subcomplexes of F P will be important later on and thus deserve some initial studies. Definition 2.1 (i-trunk) Let P be a tropical polytope and let i ∈ {1, . . . , d}. We define the i-trunk of P as This means, that we obtain Tr i (P) from P after removing every (i − 1)-tentacle element. We always have P = Tr 1 (P) ⊇ Tr 2 (P) ⊇ · · · ⊇ Tr d (P). A more general concept was introduced in [9, Def. 2.8] for arbitrary simplicial complexes, but it was not given a name there. In their notation, we have Tr i (P) = F (i,d) P . Example 2.2 shows that the 2-trunk of a 2-dimensional tropical polytope in 4dimensional space is not necessarily connected.
A particularly nice class of tropical polytopes are the pure tropical polytopes, that is, those which coincide with their d-trunk. The well-behaved nature of pure tropical polytopes was used to exhibit canonical exterior descriptions in [4]. In a similar spirit, the following statement uses a technique already occurring in the study of minimal external representations of tropical polytopes [3]. In contrast to the disconnectedness of the 2trunk in Example 2.2, it shows in particular that the d-trunk of a tropical polytope in T d is a tropical polytope itself.

Proposition 2.3
The tropical convex hull of two full-dimensional pure tropical polytopes is a pure, full-dimensional tropical polytope.
Consequently, the d-trunk of a tropical polytope in T d is a tropical polytope.
Proof Let P and Q be two full-dimensional pure tropical polytopes in T d and letP and Q be their interior. Clearly, we have tconv(P ∪ Q) ⊇ tconv(P ∪Q), where S denotes the closure in the usual topology of a set S. As P and Q are pure, we haveP = P andQ = Q. Let t = r∈R λ r r ⊕ s∈S λ s s for some finite subsets R ⊂ P, S ⊂ Q be a point in tconv(P ∪ Q) and let (r i ) i∈N → r for each r ∈ R and (s i ) i∈N → s for each s ∈ S be Together with the other inclusion, this shows tconv(P ∪ Q) = tconv(P ∪Q). For ε > 0, we define For any two points p ∈P and q ∈Q there is a sufficiently small ε > 0 such that p + B ε ⊆P and q + B ε ⊆Q. Then the 'inflated tropical line' tconv(p, q) + B ε is contained in tconv(P ∪Q). Therefore, each point is surrounded by a small full-dimensional polytrope in tconv(P ∪Q). This implies that each point of tconv(P ∪Q) is in the closure of a fulldimensional cell. Hence, tconv(P ∪ Q) = tconv(P ∪Q) is pure and full-dimensional.
The polytropes in the covector decomposition of the d-trunk are full-dimensional pure tropical polytopes P. Hence, the tropical convex hull of their union is a full-dimensional pure tropical polytope. Moreover, it is contained in the d-trunk of P, as it is a subset of P. Therefore, the tropical convex hull of the d-trunk of P is just the d-trunk itself.
The covector decomposition of a tropical polytope P = tconv(V ), where V has only integral entries, is formed of alcoved polytopes in the sense of Lam and Postnikov [32]. They studied triangulations and lattice points of alcoved polytopes from a classical point of view, while we are heading toward tropical metric estimates. Each such alcoved polytope has a triangulation into simplices of the form where π ∈ S d is a permutation of the coordinates and a ∈ Z d . For π = id, we just write (a) := id (a). We denote the simplicial complex formed by these alcoved simplices by T P and call it the alcoved triangulation of P. The inequality description of (0) is given by [7,Ch. 7]), where the all-zeroes vector is denoted by 0 = (0, . . . , 0) . We use the following notation to compactly index (half-)open faces of (0): and s (a) = a + s (0).

Tropical lattices
Recent advances on the complexity of linear programming using tropical geometry [2] demonstrated a fruitful use of metric estimates for tropical polyhedra. In classical convex geometry, the number of lattice points can be interpreted as a discrete version of a volume. This raises the question what 'tropical integers' or 'tropical natural numbers' should be.
The nonnegative integers form a submonoid of the additive monoid (R, +) generated by 1. The analogous tropical construction does not lead to a rich structure, as tropical addition is idempotent, and so 0 ⊕ 0 = 0.
Another approach comes from the property of lattices to be spanned by a finite discrete set. In particular, as lattices correspond to discrete additive subgroups of R d , they have a fix-group of translations. Although this perspective has been used in [22] and allows a tropical Ehrhart theory connected to the Euclidean volume of the polytopes in the covector decomposition (see Sect. 3), it is too rough for our purposes.
Instead, we propose to consider the set b := log b (Z ≥0 ) as a concept for tropical integers, where b ≥ 2 is an arbitrarily chosen natural number. This is natural in the sense that it respects the operation-wise transition from (R, +, ·) to the tropical semiring (R ∪ {−∞}, max, +): As additional motivation, the set b satisfies a tropicalization of the identity Our main concept of tropical lattice is therefore the following. It grasps the aspect of the lattice Z d in classical Ehrhart theory that the number of lattice points contained in a polytope suitably increases by dilation.
As we often want to vary b, we define the tropical canonical lattice TN d := (Z ≥0 ∪ {−∞}) d . Tropical b-lattices intersect exactly in TN d : Proof It suffices to prove the identity for d = 1. One inclusion is straightforward. In fact, for every m ∈ TN = Z ≥0 ∪ {−∞} and every b ∈ N ≥2 we have m = log b (b m ), with the convention that b −∞ = 0.
For the reverse inclusion, we first argue that any non-integral number in the intersection b∈N ≥2 log b (Z ≥0 ) has to be transcendental. To this end, for any b ∈ N ≥2 , by the Gelfond-Schneider Theorem (cf. [36, § 2.1]) b x is transcendental whenever x is algebraic over Z and irrational. Therefore, every x ∈ log b (Z ≥0 ) is either rational or transcendental. Assume that x = p/q is rational. We get an integer m ∈ N such that p/q = log b (m), or equivalently, m q = b p . Thus, for prime b, we must have q dividing p, and thus x = p/q is an integer. Now, if x ∈ b∈N ≥2 log b (Z ≥0 ) would be transcendental, then x, x 2 , x 3 are linearly independent over Q. Furthermore, ln(2) and ln(3) are linearly independent over Q as well, otherwise an integral power of 2 would coincide with an integral power of 3, a contradiction. Apply now the Six Exponentials Theorem (cf. [36, § 3.2]) to x i = x i , for i = 1, 2, 3, and y 1 = ln(2), y 2 = ln(3). As a result, at least one of the numbers e x i y j is transcendental. However, these exponentials are equal either to 2 x i or 3 x i , which were all assumed integers. For example, 2 x 2 = (2 x ) x is an integer using the base b = 2 x . This contradiction shows that any x ∈ b∈N ≥2 log b (Z ≥0 ) must be an integer. Tropical canonical lattice polytopes were already studied with a different motivation by Zhang [38]. They are compatible with the covector decomposition in the sense that the pseudovertices belong to TN d and to d b , for every b ∈ N ≥2 . This is however not true for (non-strong) tropical b-lattice polytopes in general, as demonstrated by the tropical 5-lattice polytope with vertices (0, 0) , (log 5 3, log 5 2) , (log 5 2, log 5 4) .

Different versions of convexity
Tropical convexity is mainly associated with the semiring S (max,+) = (R ∪ {−∞}, max, +) or, by applying the semiring isomorphism x → −x, the semiring (R ∪ {∞}, min, +). In the notation introduced before, we have T = S (max,+) . We use the latter notation whenever we need to emphasize the different semirings, and we employ the shorter and more common notation T otherwise.
While transferring from S (max,+) to S (max,·) = (R >0 ∪ {0}, max, ·) via the semiring isomorphism exp b : x → b x is often merely a structural reformulation, it has a benefit for our metric considerations, because it relates the lattice point structures over S (max,+) and S (max,·) . The next claim is far from true for general polytopes but due to the special structure of polytropes.

Proposition 2.7
The image under the map exp b of a polytrope is a polytope.  [27]. As and analogously with ≤ instead of ≥, as well as, the statement follows by taking the intersection of such sets.
More generally, the image of a weighted digraph polyhedron [28] under the exponentiation map results in a particular distributive polyhedron as studied in [19].
Consider a semiring S with addition ⊕ S and multiplication S with neutral elements 0 S and 1 S , respectively. A polytope over S is the set of finite combinations While a polytope over S (max,+) is a just a tropical polytope as defined in (2), its image under a semiring isomorphism exp b , for some b ∈ R ≥0 , is a polytope over S (max,·) . Proposition 2.7 shows that we obtain a polyhedral complex subdividing a polytope P over S (max,·) , as the image of the covector decomposition of the polytope log b (P) over S (max,+) . We call this again the covector decomposition and its vertices the pseudovertices.
A summary of the semiring isomorphisms and other involved maps is shown in Fig. 3.

Lattice point counting and semiring isomorphisms
Ehrhart's theorem on the polynomiality of the counting function k → L(P, k) of a lattice polytope P ⊆ R d , has the following powerful extension to complexes of lattice polytopes. We saw in Sect. 2.3 that a polytope over S (max,·) has a natural structure as a polyhedral complex. The appropriate lattice for the semiring S (max,·) is Z d ≥0 and thus consists of integral vectors. In analogy with classical lattice polytopes in R d , we thus call a polytope over S (max,·) a lattice polytope if all its vertices are lattice points, meaning that they belong to Z d ≥0 . We call such a polytope a strong lattice polytope if all pseudovertices of its covector decomposition are contained in Z d ≥0 . Via the isomorphism between S (max,·) and S (max,+) these notions correspond to those in Definition 2.6. We need to make this distinction for the sake of applicability of Theorem 3.1. Indeed, we get the following:

Question 3.3 How can we tell from the vertices if they span a strong lattice polytope over S (max,·) ?
Going back to tropical canonical lattice polytopes (see Definition 2.6), we actually obtain two different polynomials; one counting the lattice points in Z d , the other one counting b-lattice points. The first version is less natural from the semiring operations, but it was used in [22]. The next concept is at the heart of our quantitative studies. Definition 3.5 Let P ⊆ T d be a tropical lattice polytope and let b ∈ N ≥2 . We define the tropical lattice point enumerator of P (with respect to b) as Applying the semiring isomorphism log b to Theorem 3.2 we obtain Theorem 3.6 (Tropical Ehrhart polynomial) Let b ∈ N ≥2 and let P ⊆ T d be a tropical lattice polytope. The tropical lattice point enumerator L b P (k) agrees with a polynomial in b k for every k ∈ Z ≥0 .
Proof The set Q = exp b (P) is a strong lattice polytope over S (max,·) . Hence, by Theo- Note the use of the semiring homomorphism property of log b .

Remark 3.7
The proof above shows that the Ehrhart polynomials of P = tconv(M) ⊆ (S (max,+) ) d and Q = exp b (P) ⊆ (S (max,·) ) d agree up to a change of variables. More precisely, Remark 3.8 If one relaxes the integrality assumption in the classical setting and considers rational polytopes P ⊆ R d , that is, polytopes all of whose vertices have only rational coordinates, then their Ehrhart function k → # kP ∩ Z d turns out to be a quasi-polynomial (cf. [6,Ch. 3.8]). In the various scenarios discussed above, rationality may be defined as follows: • a polytope over S (max,·) is rational if all its pseudovertices are rational, • a polytope over S (max,+) is tropically rational if all its pseudovertices are integral (allowing possibly negative coordinates), The methods that we employed above to prove polynomiality, can similarly be used to show that in all three cases above the corresponding Ehrhart functions are quasipolynomials as well.
Definition 3.9 (Tropical Ehrhart coefficients) Let P ⊆ T d be a tropical lattice polytope. We write for its tropical Ehrhart polynomial and we call c b i (P) the ith tropical Ehrhart coefficient of P.
A very useful and reoccurring phenomenon in geometric combinatorics is reciprocity (see [7] for a detailed account). For lattice point counting functions this is known as Ehrhart-MacDonald reciprocity (cf. [6,Ch. 4]) and refers to the fact that evaluating the Ehrhart polynomial L(P, k) = d i=0 c i (P)k i of a lattice polytope P ⊆ R d at negative integers amounts to counting lattice points in the k th dilate of the interior We say that a counting function satisfying this relation fulfills reciprocity.
If a lattice polytope over S (max,·) is pure, defined analogously for polytopes over S (max,·) as over S (max,+) , the polyhedral complex induced by its covector decomposition is a dmanifold and by [33] reciprocity holds.

Explicit expressions for tropical Ehrhart coefficients
In this section, we take a much more refined route to Theorem 3.6 which is based on combining the covector decomposition with tools from classical Ehrhart theory. This allows for a refined representation of the tropical Ehrhart coefficients and leads to our desired tropical volume concept. For comparison to ordinary Ehrhart theory and further reading, we refer to [6].
In order to formulate our main technical lemma, we denote the diagonal matrix with We denote the all-one vector by 1 = (1, . . . , 1) .
Proof Clearly, φ is bijective and by definition it maps points in . As we saw above, the inequality description of the simplex s (0) is given by Here we also used that the logarithm x → log b (x) is strictly increasing.

Example 3.13
The Ehrhart polynomial of a lattice polygon P ⊆ R 2 equals a 2 ) , we use Lemma 3.11 and we get the tropical Ehrhart polynomial of (a), for each a ∈ Z 2 ≥0 : The following is our desired precise version of Theorem 3.6, building on the structure of the covector decomposition discussed in Sect. 2.1. In particular, we use the alcoved triangulation T P . It expresses the tropical Ehrhart coefficients as signed and weighted sums of the classical Ehrhart coefficients of diagonally transformed alcoved simplices.
Theorem 3.14 The ith tropical Ehrhart coefficient of the tropical lattice polytope P ⊆ T d is given by where Q denotes the closure of a set Q ⊆ R d .
Proof Every element of the alcoved triangulation T P of P, as discussed in Sect. 2.1, is of the form s π (a), for some s ∈ {=, ≤, <} d+1 and a ∈ Z ≥0 . Moreover, we think of these alcoved simplices as being relatively open, that is, s ∈ {=, <} d+1 , since this yields a partition of P into these pieces.
Therefore, the tropical lattice point enumerator is a relatively open simplex all of whose vertices lie in Z d and whose dimension is Classical Ehrhart Theory on the standard lattice Z d (cf. [6,Ch. 3]) implies that L b s π (a) (k) agrees with a polynomial in b k+1 − b k of degree m, whose coefficients depend on π, a, s, and b, but not on k. Thus, L b s π (a) (k) agrees with a polynomial in b k for every k ∈ Z ≥0 . We conclude by observing that L b P (k) as a sum of polynomials, is a polynomial in b k as well. In order to derive the stated formula for the ith tropical Ehrhart coefficient of P, we write Substituting t = (b − 1)b k and summing over all at least i-dimensional elements in T P as described above finishes the proof. Proof Let k be the dimension of T P . As for the classical Ehrhart polynomial, the highest nonvanishing coefficient is indexed by the dimension of the polytope, the right-hand side of (3) shows that c b i (P) = 0, for every i > k. Moreover, for i = k the expression (3) reduces to which is nonzero, as the sum is non-empty and every summand is positive (the highest nonvanishing Ehrhart coefficient equals the relative volume of the considered polytope; cf. [6, Sect. 5.4]).

First properties of tropical Ehrhart coefficients
Here, we record two properties of tropical Ehrhart coefficients that go well in line with their classical counterparts. We write P d T,L for the family of tropical lattice polytopes in T d . (i) (Homogeneity) For every λ ∈ Z ≥0 , we have Proof (i): We use the relationship between the Ehrhart polynomials of a tropical lattice polytope and its exp b -image in Remark 3.7. More precisely, we have the claimed identity follows by comparing coefficients.
(ii): Clearly, the counting function P → # (k P) ∩ d b is a valuation, for every fixed k ∈ Z ≥0 . Therefore, and since every involved summand is a polynomial in b k of degree d, the claim follows by comparing coefficients.  Fig. 6). We aim to compute its first tropical Ehrhart coefficient c b 1 (P). Note, that P decomposes into the alcoved triangle T = (( − 1, 0) ) and the segment S = [( , 1) , (k + , k + 1) ], which itself is decomposed into the alcoved segments S j = ( + j − 1, j) + [0, 1], for 1 ≤ j ≤ k. Hence, by the valuation property in Proposition 3.16, and the fact that the occurring intersections are zero-dimensional, we get by Lemma 3.11 and Example 3.13

A novel concept of tropical volume
Motivated by the Ehrhart polynomials from the last section, we introduce a volume notion for tropical polytopes. After stating the definition, we explain its derivation through a two stage limit process: We use the discretization of volume by lattice points and let the fineness parameter of the tropical lattice go to infinity. To get started, recall Definition 2.1 of the d-trunk as the tropical volume concept relies only on the d-trunk of a tropical polytope.  This observation also explains our choice to call tbvol(·) the tropical barycentric volume. Its tropical barycentric volume equals 0, the tropical multiplicative unit.
We now demonstrate how the tropical barycentric volume can be derived from a finer volume concept which relies on fixing the fineness parameter b of a tropical lattice. For the semiring S (max,·) , the Euclidean volume is well-behaved with respect to the arithmetic operations. As each tropical polytope P ⊆ T d is the log b -image of a polytope over S (max,·) , this motivates the following. This ties in with our deduction of tropical Ehrhart polynomials through the discretization of volume by lattice points. Using the polynomiality of the counting function k → L b P (k) established in Sect. 3, we can easily build up an analogy to the classical setting: If P ⊆ R d is a classical lattice polytope, that is, with respect to (+, ·), and with Ehrhart polynomial # kP ∩ Z d = d i=0 c i (P)k i , then by properties of the Lebesgue-measure one obtains We have k d = # k · [0, 1) d ∩ Z d , that is, k d is the number of lattice points in the kth dilate of the standard fundamental cell of Z d . The tropicalization of this statement is given by In summary, we have proved the following statement.
This example shows that the tropical b-volume of a tropical lattice polytope P equals the sum of tbvol b ( π (a)) = 1 d! (b − 1) d b a 1 +···+a d , where π (a) ∈ T P , for some a ∈ Z d and some permutation π ∈ S d . As a consequence tbvol b (P), seen as a function of b ∈ N ≥2 , is a polynomial. Hence, applying the logarithm-map to tbvol b (P), we arrive at a tropical volume concept for P which is independent of any additional parameter. The limit Log |f | does not exist for all functions f : N → R; however, we only apply it to the rational functions c b d (P) and c b d−1 (P) (cf. Lemma 5.6), which turn out to be polynomials.

Lemma 4.7 Let P ⊆ T d be a tropical lattice polytope. Then
Log |c b d (P)| = max{a 1 + · · · + a d + d : a ∈ Z d such that π (a) ∈ T P }.
Note that the alcoved simplices π (a) appearing in the latter equation are fulldimensional. Finally, we derive an expression for the limit of the leading coefficient Log |c b d (P)| of the tropical Ehrhart polynomial of P which is independent of the requirement of being a tropical lattice polytope. Proof Let P ⊆ T d be a tropical lattice polytope. The d-trunk of P is the union of all full-dimensional alcoved simplices π (a) ∈ T P . In the following, we use the compact and more convenient notation 1 x = x 1 + · · · + x d . The maximal point with respect to the linear functional 1 of such a simplex π (a) is given by a + 1, so its coordinate sum equals a 1 + · · · + a d + d = 1 a + d. The claim follows by observing that the maximal point of Tr d (P) is the maximal point of a suitable simplex π (a).
This last observation completes the two stage limit process that led us to define the tropical barycentric volume as in Definition 4.1.
With the developed notation, we can state the tropical version of the classical Pick's Theorem. It relates the volume and the number of lattice points in a lattice polygon Q ⊆ R 2 by (cf. [6, Ch. 2.6]) The symbol ∂Q denotes the boundary of the polygon Q. Similar to the tropical barycentric volume, the tropical analog of (4) is an asymptotic version of the classical one on blattices. It follows from (4) by applying the Logarithm map to Theorem 3.6 and using Proposition 4.8.

Proposition 4.9 If P ⊆ T 2 is a tropical lattice polygon, then
For a more meaningful statement, we would need to have a geometric understanding of Log |c b 1 (P)|; we refer to Sect. 5.4 for this matter.

Properties of the tropical barycentric volume
We now collect basic properties of the tropical barycentric volume, exhibiting the close analogy to the Euclidean volume. To this end, we need to introduce some notation. We write r k := r · · · r k times for tropical exponentiation. Furthermore, let P d T be the family of tropical polytopes in T d . For z ∈ T d , we consider the diagonal matrix diag(z 1 , . . . , z d ) ∈ T d×d , and for an arbitrary permutation in the symmetric group S d on d elements, let be the corresponding tropical permutation matrix. The entries in these matrices that are not specified by z ∈ T d or the corresponding permutation are −∞, the tropical zero element. The matrices of the form diag(z 1 , . . . , z d ) form the group d of scaled permutation matrices. For a matrix A = (a ij ) ∈ T d×d the tropical determinant is defined as tdet(A) = π ∈S d d i=1 a i,π (i) . The subgroup R d ⊆ d consisting of the matrices with d i=1 z i = 0, that is, those with tropical determinant equal to 0, is called the group of tropical rotation matrices. (ii) (Valuation property) tbvol : P d T → T is a valuation in the sense that for every P, Q ∈ P d T such that P ∪ Q, P ∩ Q ∈ P d T . (iii) (Rotation invariance) For M ∈ R d and P ∈ P d T , we have tbvol(M P) = tbvol(P).
We will prove a more general statement in Proposition 5.4.

Remark 4.11
Property (ii) in Proposition 4.10 actually holds in a stronger form. Indeed, tbvol : P d T → T is an idempotent measure, which means that max tbvol(P), tbvol(Q) = tbvol(P ∪ Q). For a thorough investigation of idempotent measures, we refer the reader to Akian [1].
Further, a short calculation analogous to the proof of Proposition 5.4 shows that (iii) and (iv) can be unified as tbvol(M P) = tdet(M) tbvol(P), for every M ∈ d .
The Euclidean volume vol(·) is multiplicative with respect to taking Cartesian products, that is, for any ordinary polytopes P ⊆ R d and Q ⊆ R e we have vol(P×Q) = vol(P)·vol(Q). Again, the tropical barycentric volume tbvol(·) exhibits an analogous behavior. Proof. The fact that P × Q is a tropical polytope when P and Q are, was proven in [17,Thm. 2]. The claimed identity is based on the observation that taking the trunk commutes with taking Cartesian products, more precisely Indeed, for any face F ∈ F P×Q that is contained in the (d + e)-trunk, there is a face G ∈ F P×Q with F ⊆ G and dim(G) = d + e. Since every face of a product of polytopal complexes is a product of faces of the factors, we find G P ∈ F P and G Q ∈ F Q such that G = G P × G Q , and since dim(G P ) + dim(G Q ) = d + e, we have dim(G P ) = d and dim(G Q ) = e. Therefore, writing F = F P × F Q for some F P ∈ F P and F Q ∈ F Q , we obtain F P ⊆ G P and F Q ⊆ G Q and thus F ∈ Tr d (P) × Tr d (Q). As all these arguments can be reversed, the relation (5)

Example 4.13
A tropical prism is the Cartesian product of a tropical polytope P and a 1dimensional tropical polytope L in T. As each 1-dimensional tropical polytope is pure, its tropical barycentric volume is just the coordinate of its maximal point. Writing L = [p, q], we get tbvol(P × L) = tbvol(P) + q.

Tropical volume revisited
We compare our volume notion with the two volume concepts introduced by Depersin et al. in [15].

Second highest determinant
Recall that the tropical determinant of a matrix A = (a ij ) ∈ T d×d is defined as tdet(A) = π ∈S d d i=1 a i,π (i) . Given a permutation σ ∈ S d , we further write tdet The tropical volume concept introduced in [15] can then be defined by where σ ∈ S d is a permutation at which tdet(A) is attained. Observe that this is a volume notion for matrices. For the sake of distinction, we call tvol(A) the tropical determinantal volume of A. This notion is motivated from an 'energy gap' in statistical physics used in [31]. As described in [15], the tropical determinantal volume is non-singular in the sense that tvol(A) = 0 if and only if P = tconv(A) is contained in a tropical hyperplane, and thus, if and only if tbvol(P) = −∞.
A property that distinguishes tvol(·) from tbvol(·) is that the former is translation invariant in the classical sense, that is, if we write v + A for the matrix that arises from A after adding the vector v ∈ R d to each column of A, then tvol(v + A) = tvol(A). Hence, the homogeneity of tbvol(·) described in Proposition 4.10 (iv) shows that the two volume concepts are incomparable.
Another difference with tbvol(·) is that the tropical determinantal volume is only defined for a quadratic matrix. We thus discuss potential extensions of tvol(·) to rectangular matrices. The metric quantities in Definitions 4.1 and 4.14 below are extended from a local measure to a global measure by taking a maximum, over points or submatrices. Applying this idea to tvol(·) suggests to extend it to rectangular matrices A ∈ T d×m with d ≤ m by setting where A J is the submatrix of A with columns indexed by the elements in J . This definition keeps the desirable property that the tropical determinantal volume is zero if and only if the tropical convex hull is lower-dimensional.
In the study of tropical principal component analysis, the notion tvol(·) is also discussed in [37, § 3.1]. The authors prove that the following notion also extends the tropical determinantal volume to rectangular matrices, but in terms of a sum of tropical distances: Here, is the tropical hyperplane defined by z, and is the generalized Hilbert projective metric (cf. [13]).

Tropical dequantized volume
The next concept introduced in [15] is the maximal tropical minor among the vertices. It arose from the dequantization of the Euclidean volume of polytopes over Puiseux series associated with a tropical polytope. The idea behind this formula is that the volume is essentially dominated by the maximal determinant of a simplex contained in a polytope. It turns out that the tropical dequantized volume is an upper bound on the tropical barycentric volume. This inequality is a special case of Theorem 5.12 that we prove later.  Although the previous discussion shows that the tropical volume concepts tbvol(·) and qtvol + (·) are closely related, they are inherently different. For example, the multiplicativity of tbvol(·) proved in Proposition 4.

Metric estimates for tropical polytopes
In this section, we investigate generalizations of tbvol(·) and qtvol + (·) to lowerdimensional quantities. Our definition of the tropical barycentric i-volumes below is mainly motivated by Theorem 5.7 and the discussion in Sect. 5.4, which aim to explain the second highest tropical Ehrhart coefficient as a kind of discrete tropical surface area. Another motivation comes from the connection to tropical i-minors that naturally extends Theorem 4.15 and suggests an estimate on all tropical Ehrhart coefficients (Conjecture 5.14) that has no counterpart in classical Ehrhart theory. Finally, we propose an analogy of the tropical barycentric i-volumes to the intrinsic volumes in convex geometry.

Lower-dimensional tropical volumes and their properties
We start out with our definition of lower-dimensional tropical volume measures and then derive some basic properties. respectively.
When we write tbvol ± i (·), we refer to both the upper and the lower tropical barycentric i-volume simultaneously. Each tropical barycentric i-volume comes with its own natural properties analogous to those of tbvol(·) stated in Proposition 4.10. For the rotation invariance, we need the following refined subsets of scaled permutation matrices (see Sect. 4.2): We retrieve R d = R ± d,d as a special case. We are not aware of a classical analog of R ± d,i .

Example 5.3
These subsets do not necessarily form a group for i < d as the product ⎛ Since tbvol ± d (P) = tbvol(P), the proof of the following properties also proves Proposition 4.10.

Proposition 5.4 (i) (Monotonicity) For every P, Q
(ii) (Idempotency) For every P, Q ∈ P d T such that P ∪ Q ∈ P d T , we have (iii) (Rotation invariance) For every P ∈ P d T and every M ∈ R ± d,i , we have (iv) (Homogeneity) For every λ ∈ T we have T , then Tr i (P ∪ Q) = Tr i (P) ∪ Tr i (Q) from which the claimed identity follows. (iii): The proof for tbvol + i and matrices M ∈ R + d,i is analogous. (iv): By definition Again, the proof for tbvol + i is analogous. (v): Immediate from the definition.
It is easy to check that, since Tr 1 (P) = P, we have tbvol + 1 (P) = max 1≤j≤d tbvol(π j (P)), where π j : R d → R is the projection onto the jth coordinate. This raises the question whether the tropical upper barycentric i-volumes admit a tropical analog of the integral representation formula for the intrinsic volumes (or quermassintegrals) of an ordinary polytope (see [35] for definition and properties). Roughly speaking, these formulae show that the ith intrinsic volume is the average of the volumes of the i-dimensional projections of the given polytope (cf. [10,Thm. 19.3.2] for details). However, the tropical polytope discussed in Examples 2.2 and 5.2 shows that the straightforward generalization of (10) does not hold without further reasonable assumptions.
In this line of thought, we thus pose

Question 5.5 Let P ⊆ T d be a tropical polytope. Is it true that if i ∈ [d] is an index with
Tr i (P) = P, that then where π J : R d → R |J | is the projection onto the coordinates indexed by J ?
An analogous result cannot hold for the tropical lower barycentric i-volumes. Even for i = 1, the valid inequality can be strict.

Estimating the second highest tropical Ehrhart coefficient
In this part, we argue how the tropical barycentric (d − 1)-volumes can be used to estimate the second highest tropical Ehrhart coefficient. To this end, let Q ⊆ R d be an m-dimensional classical lattice polytope, with m ≤ d. The relative volume of Q is defined as The second highest tropical Ehrhart coefficient c b d−1 (P) of a tropical lattice polytope P ⊆ T d admits a more convenient representation than the signed sum in Theorem 3.14.
To this end, recall that a simplex in T P is called maximal if it is not properly contained in another simplex of T P .
Proof Specializing Theorem 3.14 to i = d − 1 and in view of the remarks above, we have The classical description of the second highest Ehrhart coefficient of a lattice polytope (cf. (16)) implies that for d-dimensional alcoved simplices s π (a) ∈ T P , we have where the sum runs over the facets F of D a b Ğ s π (0). Each of these facets corresponds to a (d − 1)-dimensional alcoved simplex in T P , and hence, it appears in the first part of the representation of c b d−1 (P). More precisely, if the facet is contained in the interior • P of P, then it is a facet of exactly two d-dimensional alcoved simplices in T P , and so it doesn't contribute at all to c b d−1 (P). If the facet F is not contained in the interior, then it is a facet of exactly one alcoved simplex and it contributes 1 Based on this representation, we can now prove that the tropical barycentric (d − 1)volumes bound the second highest tropical Ehrhart coefficient.

Theorem 5.7 If P ⊆ T d is a tropical lattice polytope, then
Proof Our arguments are based on the representation of c b d−1 (P) given in Lemma 5.6. We start with the claimed lower bound. As a minimum of linear functions, the function attains its maximum over Tr d−1 (P) at a boundary point and thus on a (d − 1)-dimensional alcoved simplex s π (a) ∈ T P that has a nonzero contribution to c b d−1 (P). Since the boundary of the (d − 1)-trunk of P is triangulated by the closures of those s π (a), it suffices to show that for these simplices First of all, by symmetry we only need to consider π = id. In order to compute the relative volume of D a b s (0), we note that there are indices 0 ≤ j 0 < j 1 < · · · < j d−1 ≤ d such that the closure of s (0) is given by where V ∈ Z d×(d−1) is any matrix whose columns Here we used that P is a tropical lattice polytope, and thus a ∈ Z d ≥0 . Before applying (12) to estimate the determinant of said sublattice, we observe that We have, v l v k = 0, for l = k, and v l v l = j l r=j l−1 +1 b 2a r . Hence, V V is a diagonal matrix and evaluating its determinant gives the following formula that we record for later use Now, using (12) for the matrix V = (v 1 , . . . , v d−1 ), we get Putting things together we arrive at the following lower bound on the relative volume of D a b s (0): Now, the map Log |·| is monotone in the sense that Log |f | ≥ Log |g| whenever |f (b)| ≥ |g(b)| for all b ∈ N. Therefore, and so for (11) it suffices to show that The maximum on the right hand side is attained at a vertex of s (a), that is, at a point of the form a + e [j l ] , l = 1, . . . , d − 1. It thus evaluates to max l=1,...,d−1 Since j 0 < · · · < j d−1 , this implies (14) and thus the claimed lower bound on Log |c b d−1 (P)|. We now prove the upper bound. First note that the determinant of a (d −1)-dimensional sublattice L of Z d is at least 1. Indeed, there always exists a nonzero vector u ∈ Z d such that det(L) = u ≥ 1 (cf. [34,Cor. 1.3.5]). Now, let us consider an alcoved simplex s π (a) ∈ T P with dim( s π (a)) = d − 1. Again by symmetry, we can concentrate on π = id. As before, we find indices 0 ≤ j 0 < j 1 < · · · < j d−1 ≤ d such that The identity (13) yields Since Log |f + g| ≤ max{Log |f |, Log |g|}, the formula in Lemma 5.6 gives us finishing the proof.

Tropical i-minors
In this part, we aim to extend Theorem 4.15 in order to give an upper estimate for the tropical lower barycentric i-volume in terms of tropical analogs of i-minors of the defining matrix M of P. where M I,J is the i × i submatrix of M whose rows are indexed by I and whose columns are indexed by J .
For i = d, we recover the tropical dequantized volume from Sect. 4.3.2. We need a generalization of [15,Prop. 15] to all maximal tropical i-minors. In order to state it, we record that in [15] a matrix M ∈ T d×m is called tropically sign-generic if for each J ∈

Lemma 5.11
Let π ∈ S d , let 0 ≤ j 0 < j 1 < · · · < j i ≤ d be indices, and let S ∈ T d×(i+1) be the matrix whose columns are e π [j l ] := e π (1) + · · · + e π (j l ) , for l = 0, 1, . . . , i. Then, there are I ∈ [d] i and J ∈ In view of (9) and the identity tbvol(P) = Log |c b d (P)|, the following extends Theorem 4.15 to all tropical lower barycentric i-volumes. Proof. The i-trunk of P is the union of all (≥ i)-dimensional alcoved simplices occurring in the covector decomposition of P. If Tr i (P) = ∅, then tbvol − i (P) = −∞ and there is nothing to prove. We thus assume otherwise, and we let s π (a) ⊆ Tr i (P) be an alcoved simplex with dim( s π (a)) ≥ i. Of course, it suffices to show that The maximum on the left-hand side is attained at a boundary point of s π (a), so that we can assume without loss of generality that dim( s π (a)) = i. There are indices 0 ≤ j 0 < where e π [l] = e π (1) + · · · + e π (l) . Let S ∈ T d×(i+1) be the matrix whose columns correspond to the i + 1 vertices of s π (a). Combining Lemmas 5.10, 5.11, and tconv(S) = s π (a) ⊆ P = tconv(M), we see that tm i (S) ≤ tm i (M). Thus, for (15) it suffices to show To this end, we first observe that by symmetry we may assume that π = id and that j 0 = 0. Moreover, the maximum on the left hand side is attained at the point s a = a + e [j i ] , since every x ∈ s (a) is coordinate-wise dominated by s a and because the function x → v x is non-decreasing with respect to this partial order. Now, the rth coordinate of s a is given by s a r = a r + 1, if r ≤ j i , and s a r = a r , if r > j i . Therefore, s a j l = S j l ,l for every 1 ≤ l ≤ i. For i = 1 there is a more direct argument that gives a stronger result and allows to drop the integrality assumption: We conjecture that the maximal tropical i-minors also upper bound the corresponding tropical Ehrhart coefficients, and that the following analogous bound to Theorem 5.12 holds: Writing P = tconv(M), we have Thus, Log |c b 2 (P)| = 2 ≤ = tm 2 (M) and Log |c b 1 (P)| = − 1 = tm 1 (M).

Tropical surface areas
We end this section with a few musings on reasonable surface area concepts for tropical polytopes that naturally evolve from our previous studies. For one, the tropical barycentric (d − 1)-volumes may serve as surface areas. Let us thus define the upper and lower tropical surface area of a tropical polytope P ⊆ T d as tbsurf + (P) := tbvol + d−1 (P) and tbsurf − (P) := tbvol − d−1 (P), respectively. On the other hand, the second highest Ehrhart coefficient of an ordinary lattice polytope Q ⊆ R d is a kind of discrete surface area (cf. [6,Thm. 5.6]). More precisely, writing # kQ ∩ Z d = d i=0 c i (Q)k i , we have In this spirit, we may call Log |c b d−1 (P)| the discrete tropical surface area of a tropical lattice polytope P ⊆ T d . Also, the formula for c b d−1 (P) in Lemma 5.6 suggests this as a surface area concept. Natural questions for future studies arise from these definitions. First of all, we may ask for an isoperimetric inequality for tropical polytopes. The precise question taking the homogeneity of the magnitudes into account is as follows: Depersin et al. [15] established an isodiametric inequality for tropical simplices with respect to the functional tvol(·) discussed in Sect. 4.3.1 and obtained interesting families of tropical polytopes along the way. We thus ask Question 5.17 Is there an interesting isodiametric inequality with respect to tbvol(·)?
Regarding discrete surface area measures, we remark that Bey et al. [8,Prop. 4.2] proved an isoperimetric type inequality for lattice polytopes Q ⊆ R d . It states that c d−1 (Q) ≤ Kim and Roush [30,Thm. 13] showed that deciding if trk(M) ≥ k is NP-complete. Their proof shows that this is true even for 0/1-matrices and thus we conclude Theorem 6.2 Let P ⊆ T d be a tropical lattice polytope. Deciding whether max i : c b i (P) = 0 ≥ k is in general NP-hard.
Deciding whether the tropical barycentric volume tbvol(P) = Log |c b d (P)| is nonvanishing is a supposedly easier problem. For example, if P is a pure tropical lattice polytope, then by Corollary 4.17, we have tbvol(P) = qtvol + (M). In this case, the latter quantity and thus tbvol(P) can be computed in time O(m 3 ) as shown in [15]. On the other hand, this decision problem is equivalent to (a) checking non-singularity of the defining matrix M, (b) checking feasibility of a tropical linear program, and (c) deciding winning positions in mean-payoff games. All these decision problems lie in NP ∩ coNP (cf. [22, § 2.2]).

Proposition 6.3
Computing the tropical barycentric volume tbvol(P) is at least as hard as checking feasibility of a tropical linear inequality system.
One way to compute the tropical barycentric volume in Definition 4.1 is via the explicit determination of the covector decomposition, see [26], involving a classical convex hull computation.
We propose another possibility which is closer to the computation of the tropical dequantized volume defined in Definition 4.14. For this, we start by considering a tropical simplex, namely the tropical convex hull of a d × (d + 1) matrix A ∈ T d×(d+1) . We let s A ∈ T (d+1)×(d+1) arise from A by appending a zero-th row filled with tropical ones 0. With the Hungarian method, one can compute the permutation attaining the tropical determinant tdet( s A) in O(d 3 ), see [11, § 1.6.4]. Using the dual variables and reordering the columns, we can assume that the tropical determinant tdet( s A) is attained at the identity permutation, that all entries on the diagonal are 0 and that all off-diagonal entries are non-positive. One can deduce from [11,Lem. 4 As a consequence, we get that the tropical barycentric volume of a tropical polytope can be computed in polynomial time O(m d+1 ) if the dimension d is fixed, as we see in the next statement. Remark 6.6 One could consider the tropical barycentric volume tbvol(P) as a robust version of a transportation problem. The tropical dequantized volume is the generalization of a maximal matching problem, namely a transportation problem [15,Cor. 18]. The tropical barycentric volume is the solution of the transportation problem for its d-trunk, without the lower-dimensional parts. In this sense, it is more robust with respect to perturbations. Question 6.7 Let P ⊆ T d be a tropical polytope.