An optimization problem for a column, loaded by axial forces, whose direction and value remain constant, is studied in this article. The dimensional analysis introduces the dimensionless mass and rigidity factors, which simplicities the mathematical technique for the optimization problem. With the method of dimensional analysis, the solution of the nonlinear algebraic equations for the Lagrange multiplier is superfluous. The closed-form solutions for Sturm-Liouville and mixed types boundary conditions are derived. The solutions are expressed in terms of the higher transcendental function. The principal results are the closed form solution in terms of the hypergeometric and elliptic functions, the analysis of single- and bimodal regimes, and the exact bounds for the masses of the optimal columns. The proof of isoperimetric inequalities exploits the variational method and the Hölder inequality. The isoperimetric inequalities for Euler’s column are rigorously verified.