Distance matrices of subsets of the Hamming cube
Section snippets
Introduction: Distance and Gram matrices
The global geometry of a finite metric space is completely encoded within its distance matrix . A now much-studied problem concerns conditions on which ensure that embeds isometrically in a Euclidean space. We refer the reader to the surveys [2] and [10] for further details of the theory and applications of these so-called Euclidean distance matrices.
A class of finite metric spaces which rarely embeds in Euclidean space is the class of metric
Distance matrices of subsets of the Hamming cube
The hypercube has a natural additive group structure given by elementwise addition modulo 2. It is easy to check that the metric is invariant under translation in this group. In particular, if is a given subset of , then the map is an isometric isomorphism from to the set (where denotes the zero vector in ). Note that if we set , , then obviously and consequently the
Applications to supremal negative type
The results of Section 2 afford a new analysis of negative type properties of subsets of the Hamming cube . In particular, we develop a new proof of Murugan’s [11] classification of the subsets of that have strict -negative type. In order to proceed we need to recall some classical definitions and related theorems.
Definition 3.1 Let be a metric space and suppose that . Then: has -negative type iff for each finite subset of and each choice of scalars such that
Affinely independent subsets of the Hamming cube
In the proof of Murugan’s result given in the previous section, it was shown that if is an affinely independent subset of , then (where ). However, there is actually more that can be said in this setting. As stated earlier, results of Hjorth et al. [8] and Murugan [11] imply that any unweighted metric tree on vertices embeds isometrically into as an affinely independent set. For such embedded trees we may compute the precise value of
Acknowledgments
The work of the second and third authors was supported by the Research Training Program of the Department of Education and Training of the Australian Government .
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