Geometry And Combinatorics Of Tree Metrics

This paper focuses on the geometry and combinatorics of tree metrics.The main research objects are Lipschitz polytopes and polytopal complexes consisting of Lipschitz polytopes.The Lipschitz polytope of a finite metric space consists of 1-Lipschitz functions from that metric space to real numbers.Therefore,many properties of the metric space are closely related to the geometric and combinatorial properties of the Lipschitz polytope.Our research on Lipschitz polytopes proceeds in two directions: The first direction is to characterize various combinatorial properties of Lipschitz polytopes of tree metrics.This is the theme of the third chapter of this thesis,in which we give a complete combinatorial description of the Lipschitz polytope of tree metrics.On the other hand,we use the geometry of the Lipschitz polytope to study the nature of metric spaces and tree metrics.See Chapter 4 and the last section of Chapter 6 for related results.In Chapter 4 we study the Lipschitz height of tree metrics.We show that among trees with the same number of vertices,the Lipschitz height of the path is the largest and the Lipschitz height of the star tree is the smallest.In the last section of Chapter 6,we study a kind of split decomposition of metrics from the perspective of the Lipschitz polytope.Lipschitz polytope is important for the study of tropical geometry,especially for the study of tropical convex sets(tropical polytopes).Specifically,each tropical polytope has a natural polytopal complex structure(we call this complex a tropical complex),and each face in the complex is a Lipschitz polytope.We have already described in Chapter 3 when a Lipschitz polytope is a zonotope.So a natural problem is to characterize those tropical complexes whose faces are all zonotopes.A zonotopal tiling is a polytopal complex that each face is a zonotope and the union of all faces is a convex set.We prove in Chapter 5 that a tropical complex generated by a matrix is a zonotopal tiling if and only if that matrix is equivalent with the row sub-matrix of a tree metric.We also discribe the face posets of those zonotopal tilings.In Chapter 6 we study the split decomposition of tropical matrices,which are regarded as functions on the product of two simplices.It turns out that this kind of decomposition is a generalization of Buneman's decomposition on finite metrics.The results in this chapter can be used to characterize zonotopal tropical complexes.