The symmetric difference is a robust operator for measuring the error of approximating one shape by another. Given two convex shapes P and C, we study the problem of minimizing the volume of their symmetric difference under all possible scalings and translations of C. We prove that the problem can be solved by convex programming. We also present a combinatorial algorithm for convex polygons in the plane that runs in O((m+n) log^3(m+n)) expected time, where n and m denote the number of vertices of P and C, respectively.

We present an algorithm to compute an approximate overlap of two convex polytopes P_1 and P_2 in R^3 under rigid motion. Given any epsilon in (0,1/2], our algorithm runs in O(epsilon^{-3}n log^{3.5}n) time with probability 1 - n^{-O(1)} and returns a (1-epsilon)-approximate maximum overlap, provided that the maximum overlap is at least lambda max(|P_1|,|P_2|) for some given constant lambda in (0,1].