  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.
