Simultaneous inner and outer approximation of shapes

Abstract

For compact Euclidean bodiesP, Q, we define λ(P, Q) to be the smallest ratior/s wherer > 0,s > 0 satisfy\(sQ' \subseteq P \subseteq rQ''\). HeresQ denotes a scaling ofQ by the factors, andQ′,Q″ are some translates ofQ. This function λ gives us a new distance function between bodies which, unlike previously studied measures, is invariant under affine transformations. If homothetic bodies are identified, the logarithm of this function is a metric. (Two bodies arehomothetic if one can be obtained from the other by scaling and translation.)