  It is well-known that the number of non-crossing perfect matchings of $2k$
points in convex position in the plane is $C_k$, the $k$th Catalan number.
Garc\'ia, Noy, and Tejel proved in 2000 that for any set of $2k$ points in
general position, the number of such matchings is at least $C_k$. We show that
the equality holds only for sets of points in convex position, and for one
exceptional configuration of $6$ points.
