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, Gähler (Math. Nachr., 1963) (and perhaps others even earlier) defined k-metric spaces, which generalize metric spaces, and most notably generalize the triangle inequality \(d(x_1, x_2) \le d(x_1, y) + d(y, x_2)\) to the “simplex inequality” \(d(x_1, \ldots , x_k) \le \sum _{i=1}^k d(x_1, \ldots , x_{i-1}, y, x_{i+1}, \ldots , x_k)\) . (The definition holds for any fixed \(k \ge 2\) , 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 Fréchet 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.
