THE SHAPE OF THE TALLEST COLUMN

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