The shape of the tallest column

The height at which an unloaded column will buckle under its own weight is the fourth root of the least eigenvalue of a certain Sturm{Liouville operator. We show that the operator associated with the column proposed by Keller and Niordson [J. Math. Mech., 16 (1966), pp. 433-446] does not possess a discrete spectrum. This calls into question their formal use of perturbation theory, so we consider a class of designs that permits a tapered summit yet still guarantees a discrete spectrum. Within this class we prove that the least eigenvalue increases when one replaces a design with its decreasing rearrangement. This leads to a very simple proof of the existence of a tallest column.