Topological $k$-metrics

Abstract

Metric spaces $(X, d)$ are ubiquitous objects in mathematics and computer science that allow for capturing (pairwise) distance relationships $d(x, y)$ between points $x, y \in X$. Because of this, it is natural to ask what useful generalizations there are of metric spaces for capturing "$k$-wise distance relationships" $d(x_1, \ldots, x_k)$ among points $x_1, \ldots, x_k \in X$ for $k > 2$. To that end, G\"{a}hler (Math. Nachr., 1963) (and perhaps others even earlier) defined $k$-metric spaces, which generalize metric spaces, and most notably generalize the triangle inequality $d(x_1, x_2) \leq d(x_1, y) + d(y, x_2)$ to the "simplex inequality" $d(x_1, \ldots, x_k) \leq \sum_{i=1}^k d(x_1, \ldots, x_{i-1}, y, x_{i+1}, \ldots, x_k)$. (The definition holds for any fixed $k \geq 2$, and a $2$-metric space is just a (standard) metric space.) In this work, we introduce strong $k$-metric spaces, $k$-metric spaces that satisfy a topological condition stronger than the simplex inequality, which makes them "behave nicely." We also introduce coboundary $k$-metrics, which generalize $\ell_p$ 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\'{e}chet embedding (isometric embedding into $\ell_{\infty}$) and isometric embedding of all tree metrics into $\ell_1$. We also study relationships between families of (strong) $k$-metrics, and show that natural quantities, like simplex volume, are strong $k$-metrics.