We prove large deviation results for Minkowski sums of iid random compact sets where we assume that the summands have a regularly varying distribution and finite expectation. The main focus is on random convex compact sets. The results confirm the heavy-tailed large deviation heuristics: large'' values of the sum are essentially due to the largest'' summand. These results extend those in Mikosch et al. (2011) for generally non-convex sets, where we assumed that the normalization of the sum grows faster than the number of terms.