The Isoperimetric Problem in Spherical Cylinders

Abstract

The classical isoperimetric problem for volumes is solved in ℝ× n (1). Minimizers are shown to be invariant under the group O(n) acting standardly on $${\mathbb{S}}$$ n , via a symmetrization argument, and are then classified. Solutions are found among two (one-parameter) families: balls and sections of the form [a, b] × $${\mathbb{S}}$$ n . It is shown that the minimizers may be of both types. For n= 2, it is shown that the transition between the two families occurs exactly once. Some results for general n are also presented.