Matching distance

In mathematics, the matching distance is a metric on the space of size functions.

Example: The matching distance between ${\ell }_1=r+a+b$ and ${\ell }_2=r^{\prime }+a^{\prime }$ is given by $d_{\text{match}}({\ell }_1,{\ell }_2)=\max \{\delta (r,r^{\prime }),\delta (b,a^{\prime }),\delta (a,\mathrm{\Delta })\}=4$

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 ${\ell }_1$ and ${\ell }_2$, let $C_1$ (resp. $C_2$) be the multiset of all cornerpoints and cornerlines for ${\ell }_1$ (resp. ${\ell }_2$) counted with their multiplicities, augmented by adding a countable infinity of points of the diagonal $\{(x,y)\in {ℝ}^2:x=y\}$.

The matching distance between ${\ell }_1$ and ${\ell }_2$ is given by $d_{\text{match}}({\ell }_1,{\ell }_2)=\underset{\sigma }{\min }\underset{p\in C_1}{\max }\delta (p,\sigma (p))$ where $\sigma$ varies among all the bijections between $C_1$ and $C_2$ and

$\delta ((x,y),(x^{\prime },y^{\prime }))=\min \{\max \{|x-x^{\prime }|,|y-y^{\prime }|\},\max \{\frac{y-x}{2},\frac{y^{\prime }-x^{\prime }}{2}\}\}.$

Roughly speaking, the matching distance $d_{\text{match}}$ between two size functions is the minimum, over all the matchings between the cornerpoints of the two size functions, of the maximum of the $L_{\mathrm{\infty }}$-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 $\mathrm{\Delta }$. Moreover, the definition of $\delta$ implies that matching two points of the diagonal has no cost.

See also

• Size theory
• Size function
• Size functor
• Size homotopy group
• Natural pseudodistance