Set-valued metrics and generalized Hausdorff distances
Earnest Akofor
TL;DR
The paper uncovers a natural factorization of the Pompeiu–Hausdorff distance $d_H$ as $d_H=\mu\circ d_{sv}$, where $d_{sv}$ is a set-valued metric taking values in a partial algebra and $\mu$ is a postmeasure. This perspective motivates two broad families of generalized Hausdorff distances (relational and integral), unifying the Hausdorff and measure-theoretic approaches to distances between closed subsets. By developing the theory of set-valued metrics and postmeasures, the authors define sv-topologies on hyperspace subsets and construct a versatile toolkit for metric geometry of hyperspaces. The results open avenues for new hyperspace metrics, potential connections to geometric distances like Gromov–Hausdorff distances, and several open questions about sv-metrizability and the relationships among diverse ghd’s.
Abstract
Let $X$ be a (topological) space and $Cl(X)$ the collection of nonempty closed subsets of $X$. Given a topology on $Cl(X)$, making $Cl(X)$ a space, a \emph{(subset) hyperspace} of $X$ is any subspace $\mathcal{J}\subset Cl(X)$ with an embedding $X\hookrightarrow\mathcal{J}$, $x\mapsto\{x\}$ (which thus requires $X$ to be $T_1$). In this note, we highlight a key attribute of the Hausdorff distance $d_H$ on $Cl(X)$, namely, \emph{the expressibility of $d_H$ as the composition of a set-valued function and a real-valued set-function}. Using this attribute of $d_H$, we describe associated classes of distances called \emph{set-valued metrics} and \emph{generalized Hausdorff distances}.