  The mixed area of a Reuleaux polygon and its symmetric with respect to the
origin is expressed in terms of the mixed area of two explicit polygons. This
gives a geometric explanation of a classical proof due to Chakerian. Mixed
areas and volumes are also used to reformulate the minimization of the volume
under constant width constraint as isoperimetric problems. In the two
dimensional case, the equivalent formulation is solved, providing another proof
of the Blaschke-Lebesgue theorem. In the three dimensional case the proposed
relaxed formulation involves the mean width, the area and inclusion
constraints.
