The microscopic weighting on a metric space

We introduce the microscopic weighting, a canonical signed measure of mass one that can be associated to almost any finite metric space.

The microscopic weighting is obtained as the small-scale limit of the weightings used to define the magnitude function.

We give general criteria for its existence, proving in particular that every finite space of strictly negative type admits a microscopic weighting; this includes every finite subset of Euclidean or hyperbolic space and every finite tree.

Heuristically speaking, the microscopic weighting distributes its mass as widely as possible across a space, assigning greater weight to sparse or outlying regions and emphasizing points on the boundary.

Indeed, we show that on a finite space of negative type the microscopic weighting can be characterized (when it exists) as an optimizing measure for an energy integral determined by the distance function.

Alternatively, it can be characterized in terms of the geometry of the Schoenberg embedding.

Each of these interpretations also clarifies the information carried by the derivative of the magnitude function at zero.

Though our main focus in this paper is on finite metric spaces, we lay the groundwork to extend the theory to compact subsets of Euclidean space.

In that setting, we observe that the microscopic weighting must be understood as a distribution rather than as a measure.