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.

翻译：度量空间$(X, d)$是数学与计算机科学中普遍存在的对象，用于刻画点集$X$中任意两点$x, y \in X$之间的（成对）距离关系$d(x, y)$。由此自然产生的问题是：当$k > 2$时，是否存在度量空间的有用推广，用以捕捉点集$X$中$k$个点$x_1, \ldots, x_k \in X$之间的“$k$维距离关系”$d(x_1, \ldots, x_k)$？为此，Gähler (Math. Nachr., 1963)（或许更早也有其他学者）定义了$k$-度量空间，该空间推广了度量空间，特别是将三角不等式$d(x_1, x_2) \leq d(x_1, y) + d(y, x_2)$推广为“单纯形不等式”$d(x_1, \ldots, x_k) \leq \sum_{i=1}^k d(x_1, \ldots, x_{i-1}, y, x_{i+1}, \ldots, x_k)$。（该定义适用于任意固定的$k \geq 2$，且$2$-度量空间即为（标准）度量空间。）本文引入了强$k$-度量空间，这类$k$-度量空间满足比单纯形不等式更强的拓扑条件，从而具有“良好性质”。我们还引入了上边缘$k$-度量（推广了$\ell_p$度量及所有范数诱导的有限度量空间）与最小边界链$k$-度量（推广了最短路径度量且能刻画所有强$k$-度量）。利用这些定义，我们证明了有限度量空间嵌入的若干基本结果的类比版本，包括Fr échet嵌入（等距嵌入$\ell_{\infty}$）及所有树度量等距嵌入$\ell_1$。此外，我们研究了（强）$k$-度量族之间的关系，并证明单纯形体积等自然量是强$k$-度量。