Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕ
P of P that minimizes the convex hull of ϕ
P∪Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕ
P and Q to intersect or we restrict their interiors to remain disjoint; and we either allow reorienting P or require its orientation to be fixed. In the case without reorientations, we achieve exact near‐linear time algorithms for all versions of the problem. In the case with reorientations, we compute a (1+ϵ)‐approximation in time O(ϵ
-1/2log n+ϵ
-3/2log
a
(1/ϵ)) if the two sets are convex polygons with n vertices in total, where a∈{0,1,2} depending on the version of the problem.
