Tropical convexity in location problems

We investigate location problems whose optimum lies in the tropical convex hull of the input points. Firstly, we study geodesically star-convex sets under the asymmetric tropical distance and introduce the class of tropically quasiconvex functions whose sub-level sets have this shape. The latter are related to monotonic functions. Then we show that location problems whose distances are measured by tropically quasiconvex functions as before give an optimum in the tropical convex hull of the input points. We also show that a similar result holds if we replace the input points by tropically convex sets. Finally, we focus on applications to phylogenetics presenting properties of consensus methods arising from our class of location problems.


Introduction
There is a recent interest in studying location problems in tropical geometry, especially in the use of tropical methods to data analysis.Maybe the first article to promote such problems with a view towards "tropical statistics" is the work of Lin et al. [27].They showed that tropical convexity in tree spaces has some better properties than the geometry of Billera, Holmes, and Vogtmann (BHV) [7].This encouraged them to propose location estimators based on the symmetric tropical distance that could potentially exploit tropical convexity.In particular, this would give a tropical approach to the consensus problem from phylogenetics [9].
The connection for the proposed location statistics to tropical convexity was not well understood.For example, they noticed that tropical Fermat-Weber points can lie outside the tropical convex hull of the input points [27, Example 26], although it was found later that one can find Fermat-Weber points inside the tropical convex hull [34,Lemma 3.5].However, the unclear connection makes it difficult to obtain solutions that can be interpreted in the phylogenetic setting; see also [28].
Recently, we could show that studying the Fermat-Weber problem using an asymmetric distance function leads to a better explanation in terms of tropical convexity [12].In particular, it provides a clear approach based on tropical convexity to the consensus problem from phylogenetics.Moreover, various desirable properties of consensus methods were obtained by exploiting tropical convexity.In fact, the good properties were solely due to tropical convexity and not the particular distance function which motivates the search for other methods with similar properties.
In this paper, we focus on location problems that have the potential of exploiting tropical convexity.More specifically, we care of those location estimators that will belong to the tropical convex hull of the input points.Such estimators are based on distances that reflect the tropical structure of the space and can be seen as a counterpart to similar studies regarding location problems and ordinary convexity.
Significant work was done for understanding geometric properties of location problems and their relationship to ordinary convexity.The case of Chebyshev centers dates back to the 60s in the work of Garkavi [20] and Klee [25].More general location problems in a normed space were studied by Wendell and Hurter [41], while a focus on geometric properties of Fermat-Weber problems with varying distances is covered by Durier and Michelot [16].What is more, it was shown that finding an optimal solution in the (ordinary) convex hull for every set of points is equivalent to having an inner product space in three dimensions or more; a general form of this result was obtained by Durier [14,15].
The results mentioned above show a strong relationship between ordinary convexity and a Euclidean structure.Tropical convexity, on the other hand, it is related to the lattice structure of (R n , ≤).Hence, we have to focus on "monotonic" distances.To interpret geometrically monotonic functions in the quotient space R n /R1, we notice that all sub-level sets share a similarity: they are geodesically star-convex with respect to the asymmetric tropical distance.The latter can be seen by remarking that geodesic segments are images of order segments in (R n , ≤).The resulting sets, called △-star-convex, and functions, called △-star-quasiconvex, are discussed in sections 3 and 4, respectively.
In section 5 we focus on location problems in which distances to the sites are measured by △-star-quasiconvex functions.We show that this setting guarantees optimal locations in the tropical convex hull of the input.We will see that the triangle inequality does not play any role, which emphasizes the differences between tropical and ordinary convexity.Further, this setting allows for very general location problems where dissimilarities are not necessarily distances; triangle inequality is generally assumed in location science when dealing with geographic location [26], but it is not reasonable for more general data [40] and never assumed in the construction of M-estimators [23, §3.2].
We have further a few examples of location problems from the literature that end in our setting.In particular, location problems involving the symmetric and asymmetric tropical distances.However, the former case might contain cases where some optima are outside the tropical convex hull of the input.So what is the precise distinction between the symmetric and the asymmetric tropical distances that causes the above behaviour?We show that strict △-star-convexity is the answer.This motivates that study of regularized versions discussed in §5.2.
We briefly show in section 6 that we can extend the results to the case when the sites are tropically convex sets.Then section 7 deals with the main application to phylogenetics: the tropical approach to consensus methods.Our general setting provides a large class of tropically convex consensus methods as defined in [12, §5].Furthermore, we enlarge the list of desirable properties of these consensus methods that were given in the previously cited work.Finally, we conclude with section 8 consisting of highlights and possible directions for future research.

Tropical convexity
The purpose of this section is to fix the notation and emphasize the basic properties of tropically convex sets that will be used later.One can consult the book of Joswig [24] for more details.We will use both semirings T min = (R ∪ {∞}, ∧, +) and T max = (R ∪ {−∞}, ∨, +) where x ∧ y = min(x, y) and x ∨ y = max(x, y).They are isomorphic under the map x → −x, but it is better to be seen as dual to each other.This duality will play an important role later similar to the relationship between max-tropical polytopes and min-tropical hyperplanes [24,Chapter 6].
Since our applications deal with points of finite entry, we will define tropical geometric objects in R n and R n /R1.It also exploits the common set of T max and T min and we can make use of the vector space structure.
A min-tropical cone K ⊂ R n is a set closed under min-tropical linear combinations: (x + λ1) ∧ (y + µ1) ∈ K for all x, y ∈ K and λ, µ ∈ R. The image of a min-tropical cone in R n /R1 is called a min-tropically convex set.A common example is the min-tropical hyperplane with apex v which is the set The max-tropical cones and max-tropically convex sets are defined similarly, replacing min by max in the previous definitions.One can also see them as images of min-tropical cones and min-tropically convex sets under x → −x.
The min-tropical convex hull of two points a, b ∈ R n /R1 will be denoted by [a, b] max and is called the min-tropical segment between a and b.We will also use the notation The min-tropical convex hull of a set A ⊂ R n /R1 is the smallest min-tropically convex set containing A and we denote it by tconv min (A).It can be related to the max-tropical semiring by [24,Proposition 5.37].For this we need to introduce the max-tropical sector [24,Proposition 5.37] says that x belongs to tconv min (A) if and only if for each i ∈ [n] there exists a i ∈ A such that x ∈ a i + S max i .For the case of max-tropically convex hull just reverse min with max.
We say that a point a of a min-tropically convex set then we simply call it exposed.
Since the order ≤ on R n is strongly related to tropical convexity, we will focus on monotonic function.We say that a function f : X → R, defined on a subset X of R n , is increasing if for every x, y ∈ X with x ≤ y we have f (x) ≤ f (y).We call f strictly increasing if f (x) < f (y) whenever x ≤ y and x = y.
For a, b ∈ R n and a ≤ b, we denote by [a, b] ≤ the set of points x ∈ R n such that a ≤ x ≤ b and call it the order segment between a and b.It can also be written as a box: Its image in R n /R1 is a polytrope, i.e. it is both minand max-tropically convex [24, §6.5], which we call a box polytrope.A particular case is presented in the following example.
Consider the asymmetric distance d △ (a, b) = i (b i − a i ) − n min j (b j − a j ) defined on R n /R1 [12].We are interested in geodesic segments under this distance, which are portrayed in Figure 1.This is different from the geodesic convexity discussed in [24, §5.3] which focuses on the symmetric tropical distance.
) where i is any index from arg min j (b j − a j ); the equality can be also seen in Figure 1.What is more, if we choose representatives a and b such that min j (b j −a To see the latter statement, we take arbitrary representatives modulo R1 for a and b and show that a The canonical coordinates of a point x ∈ R n /R1 are the entries of the x ∈ R n defined by x = x − (min j x j ) 1.This is a representative of x modulo R1 such that all its entries are non-negative and at least one entry is 0. Definition 4. We say that K is a strictly min-tropically convex cone if K is a mintropically convex cone and for every a, b ∈ K such that a ∧ b is different from a and b modulo R1, then a ∧ b belongs to the interior of K.
We say that a subset of R n /R1 is strictly min-tropically convex if it is the image of a strictly min-tropically convex cone under the canonical projection R n → R n /R1.Proposition 6.Any strictly min-tropically convex set is a singleton or its closure coincides with the closure of its interior.Moreover, all of its boundary points are exposed.
Proof.The first part results from Remark 5.For the second part, consider v which is not exposed.Then there exist p, q in the strictly min-tropically convex set such that v ∈ (p, q) min .According to the same remark, v is an interior point.
3. △-star-convex sets Definition 7. A △-star-convex set with kernel v is a non-empty set K ⊆ R n /R1 such that for every point w ∈ K we have [v, w] △ ⊆ K.We call K strictly △-star-convex if [v, w] △ \ {w} belongs to the interior of K for every w ∈ K.
Since [v, w] △ contains the ordinary segment [v, w], we conclude that △-star-convex sets are also star-convex in the ordinary sense.We show now that △-star-convex sets are min-tropically convex.Proposition 8. Any △-star-convex set is min-tropically convex.
Proof.Let K be a △-star-convex set with kernel v and a, b arbitrary points in K.According to Remark 3, we have [a, b] The latter set is contained in K due to its △-star-convexity.
However, △-star-convex sets might not be max-tropically convex.For example, the image of the regular simplex ∆ n = conv{e 1 , . . ., e n } in R n /R1 is △-star-convex but not max-tropically convex.which is △-star-convex with kernel v-the apex.Picture (b) displays the unit balls for tropical L p norms, which will be defined in Example 14.They are nested increasingly with respect to p; the outer one corresponds to the tropical L ∞ norm and is the only one that is not strictly △-star-convex.One can recognize the triangle as the unit ball for the asymmetric tropical distance d △ .The min-tropical hyperplane with apex at the origin (the kernel of the △-star-convex sets) is dotted.
Picture (c) shows a more complicated △-star-convex sets.This case is not pure dimensional, the tropically exposed points do not form a closed set.Moreover, it is neither convex in the ordinary sense, nor strictly △-star-convex.Proposition 10.Let K be a △-star-convex set with kernel v such that K = {v}.Then K is strictly △-star-convex if and only if K is strictly min-tropically convex and v is an interior point of K.
Therefore, all of the points of [a, b] min with the exception of a and b must be in the interior of K. Hence, K is strictly min-tropically convex.The fact that v is an interior point is clear from the definition and our assumption that K = {v}.
Conversely, assume that K is strictly min-tropically convex and v is an interior point of K. We consider w ∈ K \ {v} and we show that all points of [v, w] △ \ {w} are in the interior of K.The result is clear for non-exposed points of [v, w] △ as we assumed K is strictly min-tropically convex.Hence, let u be an exposed point of [v, w] △ distinct from w.According to the discussion from Remark 2, u = w − (w j − w i )e j where i ∈ arg min k w k and j / ∈ arg min k w k .Since (u + w)/2 belongs to the interior of the tropical segment [u, w] min and K is strictly min-tropically convex, then (u + w)/2 is an interior point of K. Thus, for small δ > 0, the point c = (u + w)/2 − δe i belongs to K.
Consequently, u must be an interior point of K from the strict min-tropical convexity of K, when n ≥ 3.
For the case n = 2, we could have noticed that the exposed points of [v, w] △ are v and w, so u can only be equal to v.But v was already assumed to be interior.Remark 11.The proof above shows that the assumption that v is an interior point of K is superfluous for the converse when n ≥ 3.
But the strict △-star-convexity of K implies that a must be an interior point.
We will be interested in specific tropically quasiconvex functions.Before we introduce them, we need some notation.For a function γ : R n ≥0 → R we associate the function γ : R n /R1 → R defined by γ(x) = γ( x).We recall that x = x − (min i x i ) 1 are the canonical coordinates of x.
We will give a geometric interpretation of △-star-quasiconvex in Theorem 17.However, we prefer the definition above because it easier to check in practice.
Example 14. Considering γ a monotonic norm [6], f measures the distance to the kernel.If v = 0, then f is a gauge which are commonly used in convex analysis [36] and location science [32].Gauges are sometimes dubbed "asymmetric norms" as they satisfy all the properties of a norm with the exception that f (x) need not be equal to f (−x).
A famous class of monotonic norms are the L p norms.They give rise to △-starquasiconvex gauges whose expression is We call them tropical L p norms.They appeared in the work of Luo [29] under the name "B p -pseudonorms".One can recognize the tropical L ∞ norm as the tropical norm defined in [22, §5].The relationship to the L ∞ norm is stressed in [22, Lemma 5.2.1].The tropical L 1 norm gives rise to the asymmetric tropical distance d △ ; this relationship is implicit in [12, §6].

Remark 15. The function γ depends only on the values on ∂R n
≥0 , so we could have considered only ∂R n ≥0 as the domain of γ.However, this does not increase the generality since every (strictly) increasing function defined on ∂R n ≥0 can be extended to a (strictly) increasing function on R n ≥0 , according to the following lemma.Lemma 16.Every (strictly) increasing function γ : ∂R n ≥0 → R can be extended to a (strictly) increasing function γ : R n ≥0 → R.Moreover, if γ is continuous, then the extension can also be made continuous.
Clearly, this is continuous if γ is, as being a composition of continuous functions.Moreover, γ(x) = γ(x) for every x ∈ ∂R n ≥0 , due to monotonicity of γ and the fact that x 1 x 2 . . .x n = 0 for x ∈ ∂R n ≥0 .If x ≤ y, then x −i ≤ y −i for all i ∈ [n], where x −i is obtained from x by removing the ith entry.Therefore, γ(x −i , 0 i ) ≤ γ(y −i , 0 i ) for every i ∈ [n], which implies γ(x) ≤ γ(y) after using j x j ≤ j y j .In other words, γ is increasing.
The following result explains why the functions from Definition 13 deserve the name "△-star-quasiconvex".
Theorem 17.Let f : R n /R1 → R be a continuous function.Then f is (strictly) △-starquasiconvex if and only if all of its non-empty sub-level sets are (strictly) △-star convex with the same kernel.
Proof.After an eventual translation, we can assume that the kernel is 0.
Firstly, assume f is △-star-quasiconvex and let α ∈ R n arbitrary such that If f is strictly △-star-quasiconvex, then the points satisfying 0 ≤ x ≤ w different from w actually belong to L <α (f ).Due to the continuity of f , this coincides with the interior of L ≤α (f ).This shows that L ≤α (f ) is strictly △-star-convex.
Remark 18.The continuity of f is relevant only for strictly △-star-quasiconvex functions.Without continuity, only the strict △-star-convexity of the sub-level sets is not sufficient for f to be strictly △-star-quasiconvex.This is similar to the case of ordinary quasiconvex functions; cf.[4, Proposition 3.28] and [4,Example 3.3].
We will see that convexity, in the ordinary sense, will also be helpful for our applications.We give a simple criterion for checking when a △-star-quasiconvex function is convex.

Tropically convex location problems
We will consider some input points v 1 , . . ., v m in R n /R1.We measure the distance (or dissimilarity) from x ∈ R n /R1 to a point v i using a △-star-quasiconvex function f i having kernel v i .We consider increasing functions Without loss of generality, we assume γ i (0) = 0, so that all dissimilarities are nonnegative.
The purpose of location problems is to find a point as close (or similar) as possible to the input points, depending on some criterion; usually, the optimal location is a minimum of an objective function h : R n /R1 → R. The function h is constructed using an increasing function g : R m ≥0 → R, which aggregates the distances to the input points.Formally, we define h(x) = g (f 1 (x), . . ., f m (x)).
Since f i measures the distance or dissimilarity from x to v i and g is increasing, the minima of h record a global closeness to the input points.In most studied location problems, we would have a distance d on R n /R1 and set , for defining the Fréchet mean [18].Nevertheless, we will allow g to be an arbitrary increasing function.We will assume that h has a minimum, which happens, e.g., when h is lower semi-continuous.
Theorem 20.Let h be as above.Then there is a minimum of h belonging to tconv max (v 1 , . . ., v m ).Moreover, if g is strictly increasing and at least one of f 1 , . . ., f m is strictly △-starquasiconvex, then all the minima of h are contained in tconv max (v 1 , . . ., v m ).

Proof. Consider x /
∈ tconv max (v 1 , . . ., v m ) which is a minimum of h.Thus there exists ) for all i, and δ = min i δ i , which is strictly positive by the consideration of k.
Note that Note that the inequality above is strict if g and some γ ℓ are strictly increasing.Indeed, in that case, we must have f ℓ (x − δe k ) < f ℓ (x), so we use the strict increase of g in the ℓth entry.That would contradict the optimality of x, so the second statement of the theorem holds.
For the first statement, we can only infer that x − δe k is also a minimum of h.Hence, we can find an optimum of h in tconv max (v 1 , . . ., v m ) by moving x in directions −e k for indices k as above.
To be more precise, we collect in D(x) the possible elementary descent directions from x; formally D(x) , as the arg min functions only increase by our move in a descent direction.Thus, replacing x by x − δe k , we find a minimum with smaller D(x).We can repeat the procedure to construct a minimum x ⋆ of h with D(x ⋆ ) = ∅.The last condition is equivalent to x ⋆ ∈ tconv max (v 1 , . . ., v m ) due to [24,Proposition 5.37].
Remark 21.The regions of f i where it looks like a monotonic function are induced by the min-tropical hyperplane based at v i .Those hyperplanes defined the max-tropical polytope generated by the input points, explaining why we look at the max-tropical convex hull, instead of the min analogue.
Lemma 22. Assume that g, f 1 , . . ., f m are convex, g is strictly increasing, and at least one of the following conditions holds: a) at least one f i is strictly convex; or b) all f i are strictly convex gauges and the points v 1 , . . ., v m are not collinear.
Then h is strictly convex.In particular, it has a unique minimum.
For case b), at least one of the points v i is not on the line through x and y.Then x − v i and y − v i they are not parallel and the strict convexity of the unit ball defined by The rest of the proof is identical to case a).

5.1.
Examples.Here we review the tropical location problems from literature that fall in our category, i.e. an optimum belongs to the tropical convex hull of the input.
Example 23 (Tropical Fermat-Weber and Fréchet problems).To the best of our knowledge, the first one-point location problems in tropical geometry are proposed by Lin et al. [27].They suggest the study of Fermat-Weber points and Fréchet means under the symmetric tropical distance d trop .The goal was to relate them to tropical convexity for applications in phylogenetics.
However, they noticed that tropical Fermat-Weber points might lie outside the tropical convex hull of the input points leading to medians that cannot be interpreted easily in biological applications [27,Example 27].However, Theorem 20 says that it is possible to find an optimum in the tropical convex hull.This was already noticed for the tropical Fermat-Weber points [34, Lemma 3.5] but it was unknown, until now, for tropical Fréchet means.
Example 24 (Tropical center).Consider the case f i (x) = d △ (v i , x) and g(y) = max(y 1 , . . ., y m ).This can be interpreted as the center of the minimum max-tropical L 1 ball enclosing the points v 1 , . . ., v m .The tropical center appears in [12,Example 23], but the details are omitted.
If we choose representatives of the input points in , the optimum can be obtained by solving the linear program: (2) Note that the x-coordinates of the optimal solutions are equal, modulo R1, to the x-coordinates of the linear program (3) minimize .
Let (t ⋆ , x ⋆ ) an optimal solution of (3).For any solution of (3) we have t + x j ≥ max i∈[m] v ij =: V j .In particular, x ⋆ will have the smallest entries if we actually have equality: t ⋆ + x ⋆ j = V j , otherwise we can replace x ⋆ by some x ⋆ − εe i to minimize the objective function.This implies x ⋆ = V modulo R1; in particular, the solution is unique in R n /R1.
Even if we do not have g strictly increasing, the uniqueness and Theorem 20 ensures that the optimum is in the tropical convex hull.However, this could have been noticed from the closed form Example 25 (Transportation problems).Consider λ 1 , . . ., λ n > 0 and △(λ) the simplex in R n /R1 whose vertices are e i /λ i .Then γ △(λ) (x) = i λ i x i − ( i λ i ) min j x j is the gauge on R n /R1 whose unit ball is △(λ).
The (weighted) Fermat-Weber problem i∈[m] w i γ △(λ) (x − v i ) is equivalent to a transportation problem and every transportation problem can be reduced to this case; to see this better, write it as a linear program after scaling the weights w i such that i w i = j λ j (this change does not influence the optimum).This was firstly noticed in [12], where the authors focused on the case λ 1 = • • • = λ n .The corresponding optimum is called a tropical median in the work cited.
The optimal point is called a λ-splitter by Tokuyama and Nakano [39], but no metric interpretation was mentioned.The authors gave a condition of partitioning the space in n region in an equal fashion with some weights coming from λ and w; this can be seen as a reinterpretation of the first-order optimality condition for the corresponding Fermat-Weber problem.As a λ-splitter, it appeared in statistics [19] and as a particular case of Minkowski partition problems [3].
Example 26 (Locating tropical hyperplanes).The tropical hyperplanes are parametrized by R n /R1 by their identification with their apex.Moreover, we have d trop (a, (1) .For a vector y, we denote by y (k) the kth smallest entry, also known as the kth order statistic.Note that the aforementioned distance is △-star-quasiconvex with apex a; the easiest to see this is noticing that the second order statistic is increasing.Therefore, our general location problems cover the case of locating tropical hyperplanes.
The best-fit tropical hyperplane with with L 1 error, i.e. g is the L 1 norm, was considered by Yoshida, Zhang, and Zhang as part of tropical principal component analysis [43].
The case of L ∞ error was considered by Akian et al. [1] for applications to auction theory and called tropical linear regression.They also show that the problem is polynomialtime equivalent to mean-payoff games [1, Corollary 4.15] and, using d trop (a, H max To end this subsection, we compute the optimal location from the examples above for specific input points.We consider the points from [1, §4] which are given by the columns of the matrix (0, 0, 0) For this input, there is a unique tropical Fréchet point, (1, 1, 0), but the set of tropical Fermat-Weber points is a hexagon, marked with grey in Figure 3.We remark that V has two axes of symmetry and (1, 1, 0) is their intersection.
The point (1, 1, 0) is also the tropical center of V , while the tropical median is (0, 0, 0).The latter point is the also the unique apex of the best-fit tropical hyperplane with L 1 error of [43].It is also a solution of the tropical linear regression, but not the unique one.The apices of the best-fit tropical hyperplanes with L ∞ error are of the form (λ, λ, 0) with λ ≤ 1 and their set is pictured with green in Figure 3.

Regularization.
In some cases, we cannot expend g to be strictly increasing or all the dissimilarity functions f i to be strictly △-star-quasiconvex.Hence, a minimization algorithm might return a point outside the max-tropical convex hull of the input points, when there are multiple solutions.In this subsection, we show how we could try to arrive to a solution belonging to tconv max (v 1 , . . ., v m ) through a regularized formulation.
The idea of regularization is to consider a small parameter λ > 0 and a nicely behaved function f m+1 : R n /R1 → R ≥0 and try to solve the optimization problem minimize g (f 1 (x), . . ., f m (x)) + λf m+1 (x).
Checking more carefully the proof of Theorem 20, the second statement holds if f ℓ is strictly △-star-quasiconvex and g strictly increasing in its ℓ-th entry.We use this property for the regularization.Therefore, we obtain the following direct consequence of Theorem 20.
The influence of the term f m+1 decreases as λ goes to 0. If the functions are regular enough, we expect that a collection of optima x ⋆ λ of h λ to converge to an optimum of h.In fact, x ⋆ λ will be an optimum of h for λ sufficiently small if h is polyhedral convex and f m+1 is Lipschitz continuous.Proposition 28.If h is polyhedral convex and f m+1 is a convex function with sub-linear growth, then there exists λ 0 > 0 such that all minima of h λ are also minima of h for every λ < λ 0 .
The proof is quite technical using the differential theory from convex analysis so it is given in the appendix.We stress that Proposition 28 can be useful for studying the tropical Fermat-Weber problem from [28].Without regularization, it has undesirable behaviour for applications to biology; cf.[12, §5.2].

Location problems with tropically convex sites
Location problems can appear also when facilities are regions of the ambient space and not only points.Here, we consider such a generalization where the sites are tropically convex sets.
In the previous section, we used different distances to the input points.Here, we will measure our dissimilarities in a uniform way, by fixing an increasing function γ : R n ≥0 → R and considering d γ (x, y) = γ(y − x).We than say that d γ is △-star-quasiconvex; if γ is strictly increasing we say that d γ is strictly △-star-quasiconvex.This allows a clear definition of a distance from a region to a point: d γ (A, x) := inf y∈A d γ (y, x).
For a closed max-tropical cone K ⊆ R n we define the projection π K : R n → K as π K (x) = max{y ∈ K : y ≤ x}.We note that π K (x + λ1) = π K (x) + λ1 for every x ∈ R n and λ ∈ R, so it induces a well-defined function π K/R1 : R n /R1 → K/R1 called the tropical projection onto the max-tropically convex set K/R1.
The following lemma gives an explicit formula for the tropical projection and it characterizes it as a closest point under d γ .We omit the proof, as it is a classical result, shown when γ is the maximum norm in [10, §3] and for a general tropical L p norm in [29,Theorem 4.6].
Lemma 29.Let A be a closed max-tropically convex set.Then the tropical projection π A (x) of a point x has the entries Remark 30.In fact, the maximum expression of the tropical projection from Lemma 29 can be taken over the extremal points, in the case of tropical polytopes [24, Propositon 5.24].A similar result seems similar for general convex sets, but the form above is sufficient for our purposes.
From now on, our given sites are closed max-tropically convex sites A 1 , . . ., A m in R n /R1.Similar to section 5, the objective function is h = g (d γ (A 1 , x), . . ., d γ (A m , x)), where g : R m ≥0 → R ≥0 is increasing.Theorem 31.There exists an minimum of h lying in the tropical convex hull of the input tconv max (A 1 ∪ • • • ∪ A m ).Moreover, if g and γ are strictly increasing, then all the minima of h lie in tconv max (A [24,Proposition 5.37] entails the existence of an index ℓ ∈ [n] such that min j =ℓ (x j − a j ) < x ℓ − a ℓ for every a ∈ tconv max (A 1 ∪• • •∪A m ).Since A 1 , . . ., A m are closed sets, then there exists an open ball around x not intersecting the union of these sets.Thus, for δ > 0 sufficiently small and y = x − δe ℓ we have min j (y j −a j ) = min j (x j −a j ) for every a ∈ tconv max (A In other words, going from x in the direction −e ℓ we obtain a decrease in all the distances d γ (A i , x); in particular, a decrease of h.Using this observation, the rest of the proof is identical to the proof of Theorem 20.

Tropically convex consensus methods
In this section, we focus on applications to phylogenetics-the study of evolutionary history of species [17,37].The information is represented as an evolutionary tree, or phylogeny, which are trees whose leaves are labeled by the name of the species.In this paper, we will deal only with trees that encode the evolution from a common ancestor and possess a molecular clock.
To be more formal, we have a finite set X containing the names of the species and a rooted tree whose leaves are in bijection with X; the root corresponds to the most recent ancestor of all the species into consideration.The time is represented as positive weights on the edges, which gives a way to measure distances between nodes in the trees.What is more we assume that the distance from the root to any leaf is the same; it means that the same time is measured from the evolution of the most recent common ancestor (MRCA) of all species and any element of X.Such trees are called equidistant.
To a rooted phylogeny T we associate a distance matrix D ∈ R X×X where the entry D ij represents the distance between the leaves labelled i and j in T .It is known that T is equidistant if and only if D is ultrametric [37, Theorem 7.2.5],i.e. (5) Hence, we will not distinguish between equidistant trees and ultrametric matrices in the rest of the paper.Because D is symmetric and has zero entries on the diagonal, we can see it as a point of R ( X 2 ) .We define the tree space T X as the image of space of all ultrametrics in R ( X 2 ) /R1.Due to [2,Proposition 3], this is homeomorphic to the BHV space defined in [7].We note that the ultrametric condition (5) implies that T X is max-tropically convex.
We are interested in consensus methods: given as input multiple phylogenies on X, find an evolutionary tree on X being as similar as possible to the input trees.This is a common problem in evolutionary biology, as multiple distinct trees arise from the statistical procedures or from the multiple methods to reconstruct phylogenies from different data; see [9] or [17,Chapter 30] for details.
A consensus method can be seen as a location statistic in the tree space.Since the latter is max-tropically convex, there were many attempts to exploit this geometric structure to obtain relevant information [12,27,28,34].We are interested in tropically convex consensus methods, defined in [12].Definition 32.A consensus method c is tropically convex if c(T 1 , . . ., T m ) ∈ tconv max (T 1 , . . ., T m ) for every m ≥ 1 and T 1 , . . ., T m ∈ T X .
The location problems discussed in the previous section give rise to tropically convex consensus methods.Note that we do not need to impose the restriction that the optimum to lie in T X .It is automatically satisfied from the tropical convexity of T X and Theorem 20.This observation ensured that tropical median consensus methods are fast to compute [12, §5.3].
Tropically convex consensus methods are particularly interesting because they preserve relationships from the input trees.To explain this more clearly, we firstly need some terminology: two subsets of taxa A, B form a nesting in T , and we denote it by A < B, if the MRCA of A in T is a strict descendant of the MRCA of A ∪ B. If D is the ultrametric associated to T , then we can write the condition as (6) max i,j∈A We say that a consensus method c is Pareto on nestings if c(T 1 , . . ., T m ) displays the nesting A < B whenever A < B appears in all input trees T 1 , . . ., T m .The consensus method c is called co-Pareto on nestings if c(T 1 , . . ., T m ) does not display the nesting A < B unless A < B appears in some input tree T i .These conditions are desirable for consensus methods [9,42].
Remark 33.It is useful to see these properties from a geometric point of view.Consider T X (A < B) the subset of T X consisting of trees displaying the nesting A < B; it is described by (6).We also make the notation T X (A < B) for the complement T X \ T X (A < B), which is the set of trees not displaying A < B.
Then c is Pareto on nestings if and only if for every nesting A < B and trees T 1 , . . ., T m ∈ T X (A < B) we have c(T 1 , . . ., T m ) ∈ T X (A < B).We also note that c is co-Pareto on nestings if and only if for every nesting A < B and trees T 1 , . . ., T m ∈ T X (A < B) we have c(T 1 , . . ., T m ) ∈ T X (A < B).
The next result shows that tropically convex consensus methods have both Pareto and co-Pareto properties, being an improved version of [12,Proposition 22].Thus, we have a large class of consensus methods satisfying both properties.This is remarkable, as no such consensus method is listed in the surveys [9,42].Proof.For every nesting A < B, the set T X (A < B) is max-tropically convex as ( 6) describes an open max-tropical halfspace.Whence, Remark 33 implies that tropically convex consensus methods are Pareto on nestings.
Similarly, the set T X (A < B) is max-tropically convex as it is the intersection of T X with the tropical halfspace defined by the inequality max i,j∈A D ij ≥ max k,ℓ∈A∪B D kℓ .Remark 33 implies also the co-Pareto property.
The Pareto property gives a unanimity rule: nestings present in all the trees are also present in the consensus.One may wonder if this rule can be relaxed as there exist (super)majority-rule consensus trees commonly used for the unweighted case; they are denoted M ℓ by Felsenstein in [17, Chapter 30].Indeed, one can find such a rule for tropical medians [12].Proposition 35.A nesting appears in the tropical median consensus tree if it appears in a proportion of the input trees greater than 1 − 1/ n 2 .Moreover, a nesting will not appear in the tropical median consensus tree if it occurs in a proportion less than 1/ n 2 of the input trees.
Proof.The tropical median corresponds to the Fermat-Weber problem whose gauge distance is given by the regular simplex.Therefore, the essential hull of a finite set A defined in [13] coincides with the max-tropical convex hull of A. Then the conclusion follows from [13, Proposition 5.6] and Remark 33, as in the proof of Proposition 34.
Remark 36.Note that a consensus method is not well-defined when there are multiple minimum points.Most problematic is the situation when different tree topologies are possible, when it is unclear how to resolve incompatible optimum trees.Yet, this is not the case when the set of optimal locations is convex [11,Proposition 6]: separating the tree space in cones of trees having a tree topology gives rise to a convexly disjoint collection in the sense of [21,Definition 1.15].
Nonetheless, the aforementioned proposition applies when the set of all optima in R ( n 2 ) /R1 is contained in T X ; guaranteed for strictly △-star-quasiconvex dissimilarities.Otherwise, one might still have problems in defining consistently a consensus method; see [12,Example 24] for the symmetric tropical Fermat-Weber problem.For this reason, one has to consider the regularized versions discussed in §5.2.

Conclusion and future perspectives
We provided a large class of location estimators whose value lies in the max-tropical convex hull of the input with the purpose of obtaining consensus methods with good properties.The first direction would be to obtain methods to obtain the optima efficiently.On the other hand, searching for extra properties of specific location problems could be helpful for applications; more details are provided below.
8.1.Comparison to consensus methods based on the BHV distance.We have exploited tropical convexity to obtain consensus methods with good properties.More precisely, we focused on (co-)Pareto properties that can be interpreted in a purely geometric way.The associated spaces are also max-tropically convex so the aforesaid properties are immediate for the tropical approach.
Although the BHV geometry of the tree space is more studied than its tropical counterpart, there are few consensus methods proposed for this geometry.A first proposal was given in the pioneering paper by Billera, Holmes, and Vogtmann [7], but a few drawbacks were already pointed out: e.g., doubling every input tree changes the output.An approach based on Fréchet means was proposed by Miller et al. [31] and Bačák [5].It is also Pareto and co-Pareto on splits [31,Lemma 5.1], but the result is more intricate.The same properties hold for Fermat-Weber and center problems in the BHV space [8,Chapter 3].The approach is again analytical, but similar for all the cases.One could try a geometric approach, as in the tropical case, as it could lead faster to identification of self-consistent properties for consensus methods.8.2.Majority rules in consensus methods.Proposition 35 provides a supermajority rule for tropical median consensus with respect to nestings.This can be a step towards understanding the relationship between median weighted trees and the widely used majority-rule consensus for unweigthed trees.In fact, the majority-rule consensus can be interpreted as a median [30], but it is unclear if this can be extended to weighted phylogenies.
However, Proposition 35 provides a large threshold for a majority rule in the case of tropical median consensus trees, indicating that they are quite conservative.This seems to be owing to the low breakdown point of the tropical median caused by asymmetry; check [13] for more details.Therefore, an investigation of location estimators with higher breakdown point could provide a better connection to the majority-rule consensus.8.3.Compositional data.A different application of our location estimators could be to compositional data [35].That is, the data can be seen as points in a simplex; our methods would be applied to the centered logratio transform of the input.Note that △-star-quasiconvex sets are defined with respect to special directions, which correspond to the vertices of the simplex.
What is more, the motivation of Tokuyama and Nakano in studying algorithms for transportation problem came from splitting the points from a simplex in multiple regions [39].Moreover, Nielsen and Sun analyzed clustering methods with the symmetric tropical distance on compositional data showing a better performance than other more commonly used dissimilarity measures [33].These results suggest that △-starquasiconvex dissimilarities could be useful in compositional data analysis.
(a, b) min = [a, b] min \ {a, b} for the open min-tropical segment between a and b.Similarly, we define [a, b] max and (a, b) max .

Definition 1 .
For two points a, b ∈ R n /R1 we define the (oriented) geodesic segment between a and b under d △ as [a, b]

Remark 3 .
The set [a, b] △ contains the ordinary segment [a, b] but also the min-and maxtropical segments between a and b.What is more, for every c ∈ R n /R1 the min-tropical segment between a and b is contained in [c, a] △ ∪ [c, b] △ .

Figure 1 .
Figure 1.The geodesic segment [a, b] △ is marked with grey

Remark 5 .
A subset L of R n /R1 is strictly min-tropically convex if all the points of the open min-tropical segment (a, b) min belong to the interior of L, where a and b are distinct points in L.

vFigure 2 .
Figure 2. △-star-convex sets trop (x, H min a ), that it is dual to the problem of finding the largest inscribed ball in the tropical convex hull of the input points[1, Theorem 4.6].

Figure 3 .
Figure 3. Input points (purple) with their convex hull (black boundary) and various locations from the examples of §5.1; see the discussion after the examples

Proposition 34 .
Tropically convex consensus methods are Pareto and co-Pareto on nestings.