Isoperimetric inequality in the periodic Greenhill Problem of twisted elastic rod

In this article we demonstrate the isoperimetric inequality arising in exactly solvable structural optimization problem of stability under torque load. The Greenhill problem describes the forming of a loop in an elastic bar under torsion. The inequality for infinite rod with periodical cross-section with two types of supports is rigorously verified. The optimal shape of the twisted rod is constant along its length and the optimal shape of cross-section is the equilateral triangle. The technique to demonstrate of isoperimetric inequalities exploits the variational method and the Holder inequality.