Metric space magnitude is a numerical invariant of a finite metric space 
obtained from its similarity matrix 
, whose entries are 
. A weighting is a vector 
satisfying 
, where 
is the all-ones vector. If a weighting
exists, the magnitude is 
, a value independent of the choice of weighting.
When the similarity matrix is invertible, the weighting
is unique and
 |
This invariant can be regarded as the effective number of points in the metric space at the scale set by its metric (Leinster 2013).
A metric space is positive definite if the similarity matrix of every nonempty finite subset
is a positive definite matrix. For a nonempty
compact subset 
of such a space, magnitude can be defined as the supremum
of the magnitudes of its nonempty finite subsets (Meckes
2013).
Magnitude is nowhere continuous with respect to the Gromov-Hausdorff distance on the space of finite metric spaces
for which it is defined (Katsumasa et al. 2025). By contrast, Liu (2026) reported
that magnitude is Hausdorff distance-continuous
at every nonempty finite subset 
of a finite-dimensional positive-definite real normed
space 
.
More precisely, suppose 
has 
points, 
has dimension 
, and 
is the least eigenvalue
of 
.
Let 
be the least distance between distinct points of 
when 
, and let 
when 
. If 
satisfies 
and 
for a nonempty compact
subset 
,
then
 |
Up to linear isometry, the finite-dimensional positive-definite normed spaces are exactly the finite-dimensional
linear subspaces of 
, the Lp-space
with 
.
Liu (2026) credits GPT-6 Astra Pro and Claude Fable 5.1 with developing and refining
parts of the proof. Independent specialist review had not been reported as of Sep. 23,
2026.
Subspaces." Sep. 20, 2026.