  We study a family of norms defined for functions on an interval. These norms
are obtained by taking the $p$-norm of the Volterra operator applied to the
function. The corresponding distances have been previously studied in the
context of comparing probability measures, and special cases include the Earth
Mover's Distance and Kolmogorov Metric. We study their properties for general
signals, and show that they are robust to additive noise. We also show that the
norm-induced distance between a function and its perturbation is bounded by the
size of the perturbation, and that the distance between one-dimensional
projections of a two-dimensional function is bounded by the size of the
difference in projection directions. The results are illustrated in numerical
experiments.
