Metrics for sets of convex bodies

In many investigations into the properties of convex bodies, authors have made use of distance functions ρ( K 1 K 2 ) which give a measure of the “nearness” of two convex bodies K 1 and K 2 . Sometimes they have introduced new functions to deal with particular problems. The purpose of this paper is to compare and contrast the properties of four of these functions, namely all those (so far as we are aware) which occur in the literature and have the property that they are metrics on the set of all convex bodies of some given dimension.