A direct proof of a theorem of Blaschke and Lebesgue
  The Blaschke-Lebesgue Theorem states that among all planar convex domains of given constant width B the Reuleaux triangle has minimal area. It is the purpose of the present note to give a direct proof of this theorem by analyzing the underlying variational problem. The advantages of the proof are that it shows uniqueness (modulo rigid deformations such as rotation and translation) and leads analytically to the shape of the area-minimizing domain. Most previous proofs have relied on foreknowledge of the minimizing domain. Key parts of the analysis extend to the higher-dimensional situation, where the convex body of given constant width and minimal volume is unknown. 
Evans M. Harrell