Tropical linear spaces and tropical convexity

In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces. It is not difficult to see that each such space is tropically convex, i.e. closed under tropical linear combinations. However, we will also show that the converse is true: Each tropical variety that is also tropically convex is supported on the complex of a valuated matroid. We also prove a tropical local-to-global principle: Any closed, connected, locally tropically convex set is tropically convex.


Introduction
It has long been a consensus what the tropical analogue of a linear space should be. Sturmfels showed in [Stu02] that the tropicalization of a complex variety defined by linear equations depends only on a matroid M associated to these equations. One can give this tropical variety, the matroidal fan or Bergman fan of M in various purely combinatorial ways: E.g. through its circuits, its bases or its lattice of flats [FS05;AK06]. One can do this for any matroid, though only realizable matroids yield tropical varieties that are tropicalizations of algebraic linear spaces. In the case of fields with a nontrivial valuation, the tropicalization of a linear space is defined by a valuated matroid (M, w). This notion was originally introduced by Dress and Wenzel [DW92]. It is given by a matroid M and the additional data of a valuation w on its bases. Again, the tropical space can be defined for any such object and it was soon established that the associated tropical varieties should be called tropical linear spaces.
This terminology is further justified by the fact that being a tropical linear space is equivalent to this space having degree one: This means that it intersects the linear space of complementary dimension associated to the uniform matroid in exactly one point (where intersection is to be understood as stable intersection). A proof of this can be found in [Fin13], though the statement seems to have been known for longer (see for example [IMS07]).
Someone familiar with tropical arithmetic might expect a different definition. In the algebraic world, a linear space is simply a space that is closed under linear combinations. On the tropical side, addition and multiplication are replaced by ⊕ = max and ⊙ = +. Using these one can define tropical vector addition and tropical scalar multiplication. One might then be tempted to define a tropical linear space as a space that is closed under tropical linear combinations. This property is well-known under the name of tropical convexity. At first glance, this might seem to be a misnomer, but its justification quickly becomes clear when looking at the corresponding literature. It turns out that classical and tropical convexity are closely related. Develin and Sturmfels first introduced the concept into the tropical world [DS04] and proved -among other things -that there is a tropical Farkas' Lemma. Gaubert and Katz prove in [GK11] that tropical polytopes, i.e. the convex hulls of finitely many points, can also be written as the intersection of finitely many tropical halfspaces. Develin and Yu showed that tropical polytopes are tropicalizations of actual polytopes [DY07]. Tropical convexity has connections to many fields, such as graph theory, optimization, resolutions of monomial ideals or subdivisions of polytopes (see for example [ABGJ13; AGJ13; BY06; FR13a; JL15]).
It becomes readily apparent that simply demanding tropical convexity will not in general produce sets that are tropical linear spaces in the approved sense. However, when adding the prerequisite that the set be supported on a tropical variety, i.e. be a balanced polyhedral complex, the statement becomes true. In fact, it was already well-known that any tropical linear space (meaning a space associated to a valuated matroid) is a tropical variety supported on a tropically convex set. In this paper, we prove that the converse is also true: Theorem 1.1. Let X be a tropical variety in R n 1. Then X is tropically convex, if and only if X = B(M, w) for some valuated matroid (M, w) on [n]. In other words, X is the projectivisation of a space closed under tropical linear combinations if and only if X is supported on a tropical linear space.
In Section 2 we will review basic definitions and facts about tropical convexity, tropical varieties, valuated matroids and their associated varieties. We also include a proof of the fact that a tropical linear space is tropically convex. In Section 3 we collect results about general tropically convex complexes (i.e. that do not require balancing). We show that tropical convexity passes to recession fans and is locally preserved. We also prove that any tropically convex fan of dimension d is contained in the d-skeleton of the normal fan of the permutohedron. Section 4 then contains the actual proof of Theorem 1.1. We prove the statement first for fans and trivially valuated matroids. The general result then follows (with a bit more work) from the fact that being a tropical linear space is also equivalent to having a recession fan that is a tropical linear space. In section 5, we prove a tropical local-to-global convexity theorem: Theorem 1.2. Let X ⊆ R n 1 be a closed, connected set. If X is locally tropically convex, then X is tropically convex.
From this we deduce that being a tropical linear space is a local property.
Acknowledgement. The author was partially supported by DFG grants 4797/1-2 and JO366/3-2 1 and EPSRC grant EP/I008071/1. I would like to thank Michael Joswig for many helpful suggestions.

Preliminaries
Convention. Throughout this paper we use ⊕ = max as tropical addition. Of course, all results still hold in the min-world, one simply has to "invert" the definition of tropical linear spaces as well. We also write ⊙ = + for tropical multiplication.
A subset S of R n is called tropically convex, if for all x, y ∈ S, λ, µ ∈ R we have The tropical convex hull of a set T , denoted by tconv(T ), is the smallest tropically convex set containing T . By [DS04] it is equal to the set of all tropical linear combinations of elements in T .
Remark 2.2. It is easy to see that any tropically convex set is invariant under translation by multiples of (1, . . . , 1). Hence it is customary to consider subsets S ′ of the tropical projective torus R n 1, where 1 ∶= ⟨(1, . . . , 1)⟩. We say that such a set S ′ is tropically convex, if its preimage under the quotient map R n → R n 1 is. Note, however, that tropical arithmetic operations are not actually well-defined on R n 1.
. Let x, y ∈ R n 1. Then the tropical convex hull tconv{x, y} is of the form (⋃ k i=1 l i ), where the l i are consecutive line segments connecting x and y, whose slopes are linearly independent (0, 1)-vectors. Furthermore, the number of these line segments is k ∶= {x i − y i ; i = 1, . . . , n} − 1.
Remark 2.4. Let us make this statement more concrete: Every element x ∈ R n 1 has a well-defined heterogeneity: We can also define a partition associated to x, part(x) = I 1 ∪ ⋅ ⋅ ⋅ ∪ I het(x) of [n] ordering the entries of x descendingly. More precisely: • For any j = 1, . . . , het(x) and k, l ∈ I j , we have If we define F j ∶= ⋃ j i=1 I i for j = 1, . . . , s−1, then the tropical convex hull tconv{x, y} consists of the line segments connecting points x = p 1 , . . . , p s , where for j > 1 we set In particular p s = ∆(I s ) ⊙ y ≡ y in R n 1 and the slope of the line segment [p j , p j+1 ] is e Fj .
Example 2.5. Let n = 3. We choose a representative of each element in R 3 1 by setting the first coordinate to be 0. Choose x = (0, −1, −1), y = (0, 2, 1). Then  Definition 2.6. Let X be a pure d-dimensional rational polyhedral complex in R n 1. We will denote the support of X by X ∶= ⋃ σ∈X σ. For a cell ρ of X we define V ρ ∶= ⟨a − b; a, b ∈ ρ⟩ to be the vector space associated to the affine space spanned by ρ and we write Λ ρ ∶= V ρ ∩ Z n 1 for its lattice.
The primitive normal vector of τ with respect to σ is defined as follows: By definition there is a linear form g such that its minimal locus on σ is τ . Then there is a unique generator of Λ σ Λ τ ≅ Z, denoted by u σ τ , such that g(u σ τ ) > 0. • A tropical variety (X, ω) is a pure, rational polyhedral complex X together with a weight function ω ∶ X max → N >0 (where X max denotes the set of maximal polyhedral cells of X) fulfilling the balancing equation at each codimension one cell τ : Note that we will consider two tropical varieties to be equivalent if they have a common refinement respecting the weight functions. A tropical fan is a tropical variety whose polyhedral structure can be represented by a fan. • For a polyhedral cell σ, we denote by its recession cone. One can choose a refinement of a polyhedral complex X such that rec(X) ∶= {rec(σ); σ ∈ X} is a fan and we will call that the recession fan of X.
(see [Rau09,p. 61] for a proof of this. It also follows implicitly from [AHR14, Y = X and (assuming we have chosen a common refinement) ω Y = k ⋅ ω X for some k ∈ N. We also write this as Y = k ⋅ X. • Let X be a tropical variety and p ∈ X . By refining, we can assume without loss of generality that p is a vertex of X. We define the Star of X at p to be the fan It is easy to see that this is a balanced fan and that its support is equal to Star X (p) Figure 2. Forming the recession fan and the Star of X at a point. Note that, while rec(X) is supported on a tropical linear space, it has nontrivial weights, so Theorem 2.11 does not apply .
2.3. Tropical linear spaces. We will assume that the reader is familiar with the basic notions of matroid theory (see [Oxl11] for a comprehensive study of the topic). For a study of tropical linear spaces see for example [FR13a; Rin12; MS15; Spe08].
To quickly recap the matroid terminology we will mostly use: A circuit is a minimal dependent set and a flat is a closed set, i.e. adding any element increases the rank. Note that we will assume all matroids to be loopfree.
Dress and Wenzel [DW92] generalized the notion of a matroid to that of a valuated matroid : . . , n} together with a valuation w ∶ B → R on its set of bases B fulfilling the tropical Plücker relations: Remark 2.8. As in classical matroid theory, there are various equivalent ways of defining a valuated matroid. Another way, discovered by Murota and Tamura [MT01] is via valuated circuits. A circuit valuation is obtained by choosing a vector v C ∈ (R ∪ (−∞) 1) n for each circuit C of M such that the following are fulfilled: The paper [MT01] shows that both axiom sets are cryptomorphic. More precisely, given a valuation w ∶ B → R on the bases, a circuit valuation can be defined in the following way: Let C be a circuit. Then C is the fundamental circuit with respect to some basis B and an element i ∉ B. We set is not a basis. We will consider any valuated matroid (M, w) to be equipped with this circuit valuation.
Definition 2.9. One can define a polyhedral structure on the set and assigning weight 1 to each maximal cell we obtain a tropical variety (a proof can be found in [MS15,Theorem 4.4.5]), which we also denote by B(M, w). A tropical linear space is a tropical variety of this form.
Remark 2.10. It turns out that we essentially only need to consider trivial valuations (i.e. w ≡ 0) to prove our theorem, in which case we obtain a polyhedral fan, the matroid fan or Bergman fan of a matroid M : Note that for any valuation w on a matroid M , B(M ) = rec(B(M, w)). The key fact that will allow us to reduce the problem to matroidal fans will be the following: Theorem 2.11. Let X be a tropical variety. Then the following are equivalent: This follows from the two facts that being a tropical linear space is equivalent to having degree one [Fin13, Theorem 6.5] and that a tropical variety and its recession fan have the same degree (see for example the argument in the proof of [MS15, Theorem 4.4.5]).
It has been shown that B(M ) has several possible representations as a polyhedral fan. We will be working with the structure induced by the flats of M -this is the finest fan structure of B(M ) that usually occurs in the literature: Let F be the set of flats of M . For any chain C = (F 1 , . . . , F d = E), where ∅ ⊊ F 1 ⊊ ⋅ ⋅ ⋅ ⊊ F d = E and F i ∈ F for all i, we define a polyhedral cone: If we go through all chains of flats in M , the corresponding cones obviously form a fan and by [AK06] the support of this fan is B(M ).
Remark 2.13. One can also retrieve the matroid from its Bergman fan. It is a well-known fact that its set F of flats is {F ⊆ [n]; v F ∈ B(M )}. To see this, assume v F ∈ B(M ). By [FS05] this is the same as saying that the set of bases B of M such that B ∩ F is maximal covers all of E, as these are the bases of minimal weight with respect to v F . But that implies that F is a flat: If i ∉ F , there is a basis B containing i and having maximal intersection with F , so rank(F + i) = B ∩ (F + i) = B ∩ F + 1 = rank(F ) + 1.
The following has been known for long, but we include the proof here for completeness: Proposition 2.14. Let (M, w) be a valuated matroid of rank r on n elements. Then B(M, w) is tropically convex.
Proof. Let x, y ∈ B((M, w)) and λ, µ ∈ R. Let Let C ⊆ [n] be a circuit of M . We will denote the corresponding valuation by v C .
In that case max i∈C We will assume without restriction that Let k ∈ C be arbitrary. Then In particular, the maximum max i∈C {z i + (v C ) i } is assumed twice (at j 1 and j 2 ).

Tropically convex complexes
Proposition 3.1. Let X be a polyhedral complex and assume X is tropically convex. Then rec(X) is tropically convex as well.
Proof. Assume v, v ′ ∈ rec(X) . We can reformulate this as the fact that there exist p, p ′ ∈ X such that p It is easy to see that we can choose α large enough such that part(∆ α ) remains constant and is a refinement of part(ν), by which we mean that if ν i < ν j , this implies (∆ α ) i < (∆ α ) j . Assume we have fixed such an α and that part(∆ α ) = (I 1 , . . . , I s ). Now as in Remark 2.4 we see that the tropical convex hull tconv{q α , q ′ α } consists of line segments connecting points where F i = ⋃ k≤i I k . Now let β > α. We calculate that for all j = 1, . . . , s we have Note that r 1 , . . . , r s are exactly the vertices of the line segments forming tconv{v, v ′ } (some of them may be the same, as part(∆ α ) can be strictly finer than part(ν)).
In particular, since p β j = p α j + (β − α)r j ∈ X for any β > α, we see that r j lies in rec(X) . It is now easy to see that in fact the line segments in between must also lie in rec(X) .
Lemma 3.2. Let X be a d-dimensional polyhedral complex such that X is tropically convex. Assume that rec(X) = {rec(σ); σ ∈ X} is a fan. Then for each maximal cone ρ of rec(X), there is exactly one maximal cell σ of X such that ρ = rec(σ).
Proof. Assume there are two maximal cells σ, σ ′ of X such that ρ = rec(σ) = rec(σ ′ ). Pick any two points p ∈ σ, p ′ ∈ σ. For r ∈ ρ we write q r = p + r, q ′ r = p ′ + r. As taking the tropical convex hull commutes with translations, we see that tconv{q r , q ′ r } = tconv{p, p ′ } + r. This implies that tconv{p, p ′ } + ρ ⊆ X . But as p − p ′ ∉ V σ = V ρ , there must be a line segment l ⊆ tconv{p, p ′ }, whose slope does not lie in V σ . Hence l + ρ is a (d + 1)dimensional set contained in X , which is a contradiction to our assumption. Proposition 3.3. Let X be a tropically convex polyhedral complex and p ∈ X . Then Star X (p) is tropically convex as well.
Definition 3.4. Let X be a polyhedral fan in R n 1. We write For any chain C = (F 1 , . . . , F d = E) in F X , we will define cone(C) as in Definition 2.12. Note that each such cone is unimodular and has dimension d − 1.
We then obtain a polyhedral fan, the chain fan of X: Ch X ∶= {cone(C); C a chain in F X }.
In general this fan obviously need not be pure and can be empty.
Remark 3.5. A special case is X = R n 1. Then Ch X =∶ Ch n is a subdivision of R n 1 according to partitions, i.e. two elements lie in the interior of the same cone, if and only if they have the same partition. In particular, if x ∈ R n 1, then the minimal cone of Ch n containing x is d-dimensional if and only if het(x) = d+1. Note that Ch n can also be defined as the quotient of the normal fan of the permutohedron or as the chains-of-flats subdivision corresponding to the uniform matroid U n,n .
Lemma 3.6. Let X be a tropically convex polyhedral fan in R n 1. Then the following hold: (1) Assume dim(X) = d and x ∈ X . Then het(x) ≤ d + 1. In particular, X is contained in the d-dimensional skeleton of Ch n .
(3) If C is any chain in F X , then cone(C) ⊆ X . Proof.
(1) Fix a representative of x and assume that s ∶= het(x) > d + 1. Let part(x) = I 1 ∪ ⋅ ⋅ ⋅ ∪ I s as in Remark 2.4 and F j ∶= ⋃ i≤j I i . The tropical line segment tconv{0, x} consists of segments connecting the points 0 = p 1 , . . . , p s = x, where We will show inductively that for k = 2, . . . , s − 1 there are continuous, concave, piecewise affine linear functions ε k ∶ R k−1 → R, such that ε k is strictly positive on (R >0 ) k−1 and such that for k ≥ 1 we have Then P s−1 is a polyhedron of dimension s − 1 > d in the fan X, which is a contradiction to our assumption that dim X = d.
For k = 1 we get P 1 = {λ 1 e I1 ; λ 1 ≥ 0}. As p 2 = (x(I 1 ) − x(I 2 ))e I1 ∈ X and X is a fan, this is always contained in X. Now assume k > 1 and that we have found ε 2 , . . . , ε k−1 . Let q = ∑ k−1 i=1 λ i e Ii ∈ P k−1 and assume 0 < λ i for all i. In particular, q(I i ) > 0 for all i < k and q(I i ) = 0 for i ≥ k. We now consider the tropical line segment tconv{αq, p k+1 }, where α > 0. By induction q ∈ X and since X is a fan, so is αq. We conclude that tconv{αq, p k+1 } ⊆ X . Let ∆ = p k+1 − αq. Note that ∆ has constant entries on each I j (as both q and p k+1 do). More precisely, we have Now, if we pick α sufficiently large, ∆ is maximal on entries in I k : For j ≠ k we have , if j < k and the latter is always greater than zero if we pick In this case the first line segment of tconv{αq, p k+1 }, going from αq to p k+1 , has slope e I k . Its length is d k ∶= x(I k ) − x(I k+1 ): Our choice of α implies that ∆(I j ) < 0 for all j < k.
As ε k clearly fulfills all desired properties, the claim follows.

Tropical convexity and valuated matroids
By Proposition 2.14 we only need to prove that any tropically convex tropical variety is supported on a tropical linear space. In fact, it will suffice to reduce to the case of fans: Proposition 4.1. Let X be a tropical variety whose support is a tropically convex fan. Then X = B(M ) for a matroid M .
Our main theorem now follows from this: Proof. (of Theorem 1.1) The "if" direction follows from Proposition 2.14. For the "only if" direction, let X be a tropical variety with tropically convex support. By Propositions 3.3 and 4.1, Star X (p) is supported on a matroidal fan B(M (p)) at each p ∈ X . Since any matroidal fan is irreducible [FR13b, Lemma 2.4], we must have Star X (p) = k p ⋅ B(M (p)) for some k p ∈ N. As X is tropically convex, it is also path-connected. This implies that k p = k p ′ =∶ k for all p, p ′ ∈ X . We conclude that X = k ⋅ Y for some k ∈ N and a tropical variety Y with constant weight 1.
By Propositions 3.1 and 4.1, rec(X) is also supported on a matroidal fan and hence of the form l ⋅ B(M ) for some l ∈ N and some matroid M . Using Lemma 3.2, we see that rec(Y ) = B(M ) (and in fact: k = l). By Theorem 2.11 we must have Y = B(M, w) for some valuation w on M , so X = Y = B(M, w) , as claimed.
The general idea for proving Proposition 4.1 is to revert the procedure described in Remark 2.13: We define M via its flats, which is the set F X of all sets whose incidence vectors lie in X . We show that X is supported on the fan of chains of F X . Then it only remains to show that F X actually fulfills the axioms required for a set of flats. Naturally, the balancing condition plays a crucial role in the proofs of both statements.
Proposition 4.2. Let X be a tropical fan and assume X is tropically convex. Then X = Ch X .
Proof. Let d ∶= dim X and let X be a polyhedral structure of X. By Lemma 3.6,(1), X is contained in the d-dimensional skeleton of Ch n . By intersecting X with Ch n , we can now assume that each cone of X is contained in some d-dimensional cone(C), where C is a chain of arbitrary subsets of [n].
The balancing condition of X now dictates that if ρ = cone(C) ∈ Ch n contains a maximal cone of X in its interior, all of ρ must be in X : Otherwise, it would contain a codimension one face τ of X in its interior, such that there is one maximal cone σ > τ with σ ⊆ ρ but no other maximal cone σ ′ > τ is contained in ρ. Balancing implies that at least one other maximal cone σ ′ is adjacent to τ . By our previous argument, σ and σ ′ both lie in d-dimensional cones of Ch n . But these cones intersect only in their boundary.
This proves X ⊆ Ch X and the converse follows from Lemma 3.6,(3).
Proof. (of Proposition 4.1) Proposition 4.2 tells us that we can equip X with the polyhedral structure of Ch X : Balancing implies that all cones contained in a cone of Ch X must have the same weight. Hence it is sufficient to show that F X defines indeed a set of flats of a matroid. More precisely, we have to show the following: (3) Let F ∈ F X and assume F 1 , . . . , F k are the minimal elements of The first statement is trivial and the second follows from Lemma 3.6, (2).
To prove the third axiom, let F ∈ F X and denote by F 1 , . . . , F k the minimal elements of F X containing it. Then (F i ∖ F ) ∩ (F j ∖ F ) = ∅ by minimality and the second axiom. Hence we only have to prove that ⋃ k i=1 F i = E. By Proposition 4.2 and the fact that X is pure of some dimension d, every maximal chain in F X has the same length d + 1. Let G ∈ F X and C = (F 1 , . . . , F d+1 = E) a maximal chain in F X . We define the rank of G to be rank(G) ∶= i, if F i = G. This is independent of the actual chain: Otherwise we could combine two chains with G occurring at different positions to form a chain of length greater than d + 1.
We will now prove the last axiom by induction on c(F ) ∶= d + 1 − rank(F ). If c(F ) = 1, then k = 1 and F 1 = E, so the statement is true. Now let c(F ) > 1 and j ∈ E ∖ F . By induction, there exists an F ′ ∈ F X of rank rank(F ) + 2 and with F ⊆ F ′ , such that j ∈ F ′ . We can now pick a chain where rank(G i ) = i for all i (such a chain exists, as X is pure). Then cone(D) is a codimension one cone. Using the fact that all chain cones are unimodular, the balancing equation at cone(D) reads: v ∶= . . , F k } and w G ∈ Z ∖ {0} denotes the weight of the corresponding maximal cone.
As v ∈ V cone(D) , all entries {v i , i ∈ F ′ ∖F } agree (this notion is obviously well-defined in R n 1). If we pick the obvious representative − ∑ i∈G e i ∈ R n for each v G and use the fact that for any F s , F t ∈ G(D) we have (F s ∖F )∩(F t ∖F ) = ∅, then for i ∈ F ′ ∖F we have As X is pure, G(D) is not empty. But this implies that there must be an F s ∈ G(D) with j ∈ F s .

Local-to-global tropical convexity
In classical convexity theory, there are various local-to-global principles. In this section, we prove Theorem 1.2, a tropical analogue of a result proven by Tietze and Nakajima [Tie28;Nak28]. It states that any closed connected subset of R n , which is locally convex, is already convex. The main strategy of the proof follows a standard argument for classical convexity -though there is some extra work involved due to the fact that R n 1 is not uniquely geodesic with its canonical metric (see Remark 5.3).
Definition 5.1. Let x ∈ R n 1. We define the tropical norm of x to be We also fix the following notations: For a compact set S and a point x, we will also write The following all have easy and elementary proofs: Lemma 5.2.
(1) ⋅ trop is twice the quotient norm of the maximum norm on R n . In particular, it defines a norm on the R-vector space R n 1. (4) If x ∈ R n 1 and tconv{0, x} consists of actual line segments connecting points 0 = p 1 , . . . , p s = x, then (1, 0, 1) (0, 0, 1) (0, 1, 1) (0, 1, 0) (1, 1, 0) (1, 0, 0) Remark 5.3. The tropical norm was already introduced in [DS04] to study tree metrics. Joswig shows that R n 1 is a geodesic space [Jos]: The tropical line segment between two points is a geodesic with respect to the metric induced by ⋅ trop . However, it is not uniquely geodesic. There are generally various paths from x to y whose length is x − y trop (see also Figure 4).
x y Figure 4. The set of points z with x − z trop = z − y trop = x − y trop 2 is a polytrope.
Lemma 5.4. Let x, y, z ∈ R n 1. Then for any point p ∈ tconv{x, y}, we have Proof. Let j ∈ I max (p − z) and assume j ∉ I max (x − z). We know that we can write y = x+∑ s i=1 α i e Fi , where α i > 0 and F i ⊊ F i+1 for all i and each summand corresponds to a vertex on the tropical line segment tconv{x, y}. Hence p = x+∑ k−1 i=1 α i e Fi +βe F k for some k ≤ s and β ≤ α k . Since p j − z j = (x j − z j ) + ∑ k−1 i=1 α i (e Fi ) j + β(e F k ) j is maximal and x j − z j is not maximal, we must have that j ∈ F m for some m ≤ k. In particular, j ∈ F l for all l ≥ k, so y j − z j is still maximal.
Definition 5.5. Let X ⊆ R n 1 and x, y ∈ X.
• We call X locally tropically convex, if for every x ∈ X there exists an ε > 0, such that B trop ε (x) ∩ X is tropically convex. • A tropical path in X from x to y is an injective continuous map γ ∶ [0, 1] → X, whose image is a concatenation of tropical line segments leading from x to y. The length l(γ) of γ is the length with respect to ⋅ trop , i.e. if γ consists of tropical line segments connecting x = x 0 , . . . , x k = y, then • We define the distance of x and y in X to be d X (x, y) ∶= inf{l(γ); γ a tropical path from x to y.}.
Lemma 5.6. Let X ⊆ R n 1 be locally tropically convex and x, y ∈ X. Assume there is a point z ∈ X such that the following hold: • d X (x, z) = d X (z, y) = d X (x, y) 2.
• z −tconv{x, y} trop is minimal among all points fulfilling the first property.
Proof. Assume z ∉ tconv{x, y}. We define F ∶= I max (x − z), F ′ ∶= I max (y − z). Then e F , e F ′ are the outgoing slopes of the tropical line segments from z to x and y, respectively (see also Figure 5 for an illustration). Now choose ε > 0 small and let z ′ ∶= (z + εe F ) ⊕ (z + εe F ′ ) = z + εe F ∪F ′ .
By local tropical convexity, this lies in X for sufficiently small ε.
First of all, we see that z ′ still fulfills the first property: Note that the concatenation of tconv{x, z + εe F } and tconv{z + εe F , z ′ } forms a tropical path in X from x to z ′ . Again assuming ε to be sufficiently small and using that x ≠ z, we get z). Proof. (of Theorem 1.2) Let x, y ∈ X and set r ∶= d X (x, y).
Since X is closed and locally tropically convex, there exists a midpoint, i.e. a point z ∈ X such that d X (x, z) = d X (y, z) = 1 2 r.
Note that the set of midpoints of x and y is a compact set. It is obviously closed and must be a subset of B trop r 2 (x) ∪ B trop r 2 (y). Hence we can choose z to have minimal distance to tconv{x, y}.
In this manner, we recursively construct points z i,n ∈ X with 1 ≤ n, 0 ≤ i ≤ 2 n , such that • z 0,n = x, z n,n = y and z i,n = z 2i,n+1 .
• z 2i+1,n+1 is a midpoint of z i,n and z i+1,n and it has minimal distance to tconv{z i,n , z i+1,n }.
In particular, we have d X (z i,n , z i+1,n ) = r 2 n . Now we have z i,n − x trop ≤ d X (z i,n , x) ≤ (i 2 m )r ≤ r for all i and n, so z i,n ∈ B trop r (x) ∩ X =∶ B, which is a compact set. Hence we can choose a global δ > 0 such that for all x ∈ B, the set B trop δ (x) ∩ X is tropically convex.
By choosing n large enough, we can now assume that r 2 n+1 < δ. Then for each i, B trop δ (z 2i+1,n+1 ) contains both z i,n and z i+1,n , so their tropical convex hull is contained in X. Applying Lemma 5.6 inductively, we see that tconv{x, y} ⊆ X.
Corollary 5.7. Let X be a connected tropical variety in R n 1, which is locally a multiple of a matroidal fan, i.e. Star X (p) = k p ⋅ B(M (p)) for each p ∈ X, some k p ∈ Z and some matroid M (p). Then X is supported on a tropical linear space.