Multidimensional persistence studies topological features of shapes by
analyzing the lower level sets of vector-valued functions. The rank invariant
completely determines the multidimensional analogue of persistent homology
groups. We prove that multidimensional rank invariants are stable with respect
to function perturbations. More precisely, we construct a distance between rank
invariants such that small changes of the function imply only small changes of
the rank invariant. This result can be obtained by assuming the function to be
just continuous. Multidimensional stability opens the way to a stable shape
comparison methodology based on multidimensional persistence.Comment: 14 pages, 3 figure