Mixed volumes and the Blaschke-Lebesgue theorem

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.