  This paper studies the properties of a new lower bound for the natural
pseudo-distance. The natural pseudo-distance is a dissimilarity measure between
shapes, where a shape is viewed as a topological space endowed with a
real-valued continuous function. Measuring dissimilarity amounts to minimizing
the change in the functions due to the application of homeomorphisms between
topological spaces, with respect to the $L_\infty$-norm. In order to obtain the
lower bound, a suitable metric between size functions, called matching
distance, is introduced. It compares size functions by solving an optimal
matching problem between countable point sets. The matching distance is shown
to be resistant to perturbations, implying that it is always smaller than the
natural pseudo-distance. We also prove that the lower bound so obtained is
sharp and cannot be improved by any other distance between size functions.
