The problem of optimizing the torsional rigidity of a prismatic body about a cross section

Some extremal problems, related to the torsional rigidity of a homogeneous body, are investigated. The problem of optimizing the torsional rigidity of a cylindrical body about a cross section is solved by determining the variation of the region when using the one-to-one correspondence between bounded convex regions and continuous positive-homogeneous convex functions. Using this approach a formula is obtained for the torsional rigidity, and conditions are also obtained characterizing the optimum region and the maximum value of the functional.