Distances between fuzzy sets

Given two fuzzy subsets @m and @n of a metric space S (e.g., the Euclidean plane), we define the 'shortest distance' between @m and @n as a density function on the non-negative reals; our definition is applicable both when @m and @n are discrete-valued and when they are 'smooth' (i.e., differentiable), and it generalizes the definition of shortest distance for crisp sets in a natural way. We also define the mean distance between @m and @n, and show how it relates to the shortest distance. the relationship to earlier definitions of distance between fuzzy sets [1,3] is also discussed.