# Matching distance

In mathematics, the **matching distance**^{[1]}^{[2]} is a metric on the space of size functions.

The core of the definition of matching distance is the observation that the
information contained in a size function can be combinatorially stored in a formal series of lines and points of the plane, called respectively *cornerlines* and *cornerpoints*.

Given two size functions and , let (resp. ) be the multiset of all cornerpoints and cornerlines for (resp. ) counted with their multiplicities, augmented by adding a countable infinity of points of the diagonal .

The *matching distance* between and is given by
where varies among all the bijections between and and

Roughly speaking, the matching distance between two size functions is the minimum, over all the matchings between the cornerpoints of the two size functions, of the maximum of the -distances between two matched cornerpoints. Since two size functions can have a different number of cornerpoints, these can be also matched to points of the diagonal . Moreover, the definition of implies that matching two points of the diagonal has no cost.

## References

- ↑ Michele d'Amico, Patrizio Frosini, Claudia Landi,
*Using matching distance in Size Theory: a survey*, International Journal of Imaging Systems and Technology, 16(5):154–161, 2006. - ↑ Michele d'Amico, Patrizio Frosini, Claudia Landi,
*Natural pseudo-distance and optimal matching between reduced size functions*, Acta Applicandae Mathematicae, 109(2):527-554, 2010.