Page 32 - Tequio 11

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

where - is the unit vector with entry one in the first coordinate and zeros otherwise, i.e., P is the vector with
the only nonzero entry one in the -th coordinate.
3

Figure 3.
A square (β2) and an octahedron (β 3) with f-vectors (4; 4) and (6; 8; 8).

Another interesting class of polytopes are zonotopes, which are Minkowski sums of lines. Their combinatorial
structure connects to hyperplane arrangements, tilings or oriented matroids (Ziegler, 1995, x 7). As an example
consider the square of figure 3 as the sum of the lines [ - , * ], [ - , − * ].

2.3 Finite metric spaces and splits
Let be a set. A metric (or distance function) on is a symmetric function : × → ℝ n, such that
(1) For all , ∈ , (, ) = 0 implies = .
(2) For all , , ∈ , (, ) ≤ (, ) + (, ) (“triangle inequality”).
If condition (1) is dropped, then is called a pseudometric. In the following we will focus on finite metric
spaces with || < ∞.

Example 2.3 (Metric spaces from weighted graphs) A weighting of a graph is any function : () →
ℝ v, , and the pair (, ) is called a weighted graph. Set

w (, ′): = min{( - ) + ⋯ + ( y ) ∣ , - , - … , y , ′ is a path joining with ′}

such that the pair ((), w ) is a metric space.

If (, ) represents (, ) it is called a graph realisation of the metric space. Note that any finite metric space
4
has a graph realisation from the complete graph by setting the weight of the edge P,Ä between , to (, ).
Next, we introduce metric spaces coming from -trees.

3 I.e. i stands for any of the elements i∈{1,⋯,d}.
3
I.e. stands for any of the elements ∈ {1, ⋯ , }.
4 The complete graph on a set of vertices is the graph where any two vertices are connected to each other through an edge.
4 The complete graph on a set of vertices is the graph where any two vertices are connected to each other through an edge.
Tequio, enero-abril 2021, vol. 4, no. 11