Graduate Thesis Or Dissertation
 

On a Topological Criterion for the Construction of Higher Arity Metrics

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/mw22vf172

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  • Metric spaces (X, d) are ubiquitous objects in mathematics and computer science that are able to capture pairwise distance relationships d(x, y) between points x, y ∈ X. Because of this, it is natural to ask what useful generalizations there are of metric spaces for capturing “k-wise distance relationships” d(x1, . . . , xk) among points x1, . . . , xk ∈ X for k > 2. To that end, G ̈ahler (Math. Nachr., 1963) defined k-metric spaces, which generalize metric spaces, and most notably generalize the triangle inequality d(x1, x2) ≤ d(x1, y) + d(y, x2) to the “simplex inequality” d(x1, . . . , xk) ≤ ∑d(x1, . . . , xi−1, y, xi+1, . . . , xk). In this work, we introduce strong k-metric spaces, k-metric spaces that satisfy a topolog- ical condition equivalent to the triangle inequality for k = 2, but stronger than the simplex inequality for k > 2, which makes them “behave nicely.” We also introduce coboundary k-metrics, which generalize ℓp metrics (and in fact all finite metric spaces induced by norms) and minimum bounding chain k-metrics, which generalize shortest path metrics (and capture all strong k-metrics). Using these definitions, we prove analogs of a number of fundamental results about embedding finite metric spaces including Fr ́echet embedding (isometric embed- ding into ℓ∞). We also study relationships between families of (strong) k-metrics, and show that natural quantities, like simplex volume, are strong k-metrics
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