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On some polytopes in phylogenetics/Hoessly/27-40 35

Theorem 3.4 (Dress, 1984, Theorem 8) The metric space (, ) is tree-like if and only if the tight span (ô,j) is
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an ℝ-tree.
This has been generalised to show that for (, ) a finite metric that is totally decomposable, the Buneman
graph representing the split decomposition is contained in the 1-skeleton of the tight span (ô,j) (cf., i.e.,
(Huson et al., 2010, x 5.12)). Injectivity of metrics corresponds to some factorisation property, where Isbell
showed that ( (ô,j) , || ⋅ || û ) is injective and that every metric can be isometrically embedded into this metric
space (Isbell, 1964).

3.2 Lipschitz polytope
Studying fundamental polytopes was proposed by Vershik (Vershik, 2015) as an approach to a combinatorial
classification of metric spaces. It also relates to an isometric embedding of the metric space, however, through
optimal transport. As in the case of the Tight span it can be expected that properties of metric spaces can be
connected to properties of the fundamental polytope.
The polar dual of the fundamental polytope consists of the real-valued functions with Lipschitz constant at
most 1, called Lipschitz polytope. As polar duality preserves all combinatorial data, it is enough to classify the
combinatorial structure of Lipschitz polytopes. We will mostly focus on Lipschitz polytopes in the following.
The structure of fundamental polytopes of tree-like metric spaces were studied via associated hyperplane
arrangements and corresponding decompositions of the matroid in (Delucchi & Hoessly, 2020), enabling explicit
formulas for face numbers of tree-like finite metric spaces. Values of the -vectors as well as concrete values
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for -vectors for “generic” metrics were given in (Gordon & Petrov, 2017). For more on connections,
terminology, history and further context around fundamental polytopes we refer to, e.g., (Ostrovska &
Ostrovskii, 2019, x 1.6) or (Delucchi & Hoessly, 2020), where we further remark that direct applications to
phylogenetics are still outstanding.

Definition 3.5 The Lipschitz polytope of (, ) is given as an intersection of halfspaces by

Next we concentrate on the case of tree-like metric spaces and their Lipschitz polytopes as in (Delucchi &
Hoessly, 2020). Let be a finite set and consider a split = | of , where || = . To we associate the

line segment (one-dimensional polytope)
:::
where Accordingly, associated to a split system we define the zonotope defined by the Minkowski sum

(): = ∑ ã∈ ã .

10 An ℝ-tree (also called real trees) in some ℝ corresponds to the points of a graph-theoretical embedding of a tree.
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11 An R-tree (also called real trees) in some R^n corresponds to the points of a graph-theoretical embedding of a tree.
In (Gordon & Petrov, 2017), finite metric spaces are called generic if the triangle inequality is always strict and the fundamental
In (Gordon & Petrov, 2017), finite metric spaces are called generic if the triangle inequality is always strict and the fundamental polytope is simplicial.
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polytope is simplicial.
Tequio, enero-abril 2021, vol. 4, no. 11