<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume10 Issue1" %>
Dimensions of Tight Spans
Mike Develin
American Institute of Mathematics, 360 Portage Ave., Palo Alto, CA 94306-2244, USA
Annals of Combinatorics 10 (1) p.53-61 March, 2006
AMS Subject Classification: 51K05, 05C12, 52B45
Given a finite metric, one can construct its tight span, a geometric object representing the metric. The dimension of a tight span encodes, among other things, the size of the space of explanatory trees for that metric; for instance, if the metric is a tree metric, the dimension of the tight span is one. We show that the dimension of the tight span of a generic metric is between and , and that both bounds are tight.

Keywords: tight spans, finite metrics, geometric representation, tree metrics


1. A. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups, Adv. Math. 53 (1984) 321--402.

2. A. Dress, K.T. Huber, and V. Moulton, An explicit computation of the injective hull of certain finite metric spaces in terms of their associated Buneman complex, Adv. Math. 168 (2002) 1--28.

3. J. Isbell, Six theorems about metric spaces, Comment. Math. Helv. 39 (1964) 65--74.

4. A. Schrijver, Polyhedra and efficiency, In: Combinatorial Optimization, Vol. A, Springer, Berlin, 2003.

5. B. Sturmfels and J. Yu, Classification of six-point metrics, arXiv:math.MG/0403147, to appear.

6. G. Ziegler, Lectures on Polytopes, Springer, New York, 1995.