The isoperimetric problem in spherical cylinders (Q1771561)

A region \(\Omega\) minimizing \({H}^{n-1}(\partial\Omega)\) under a volume constraint is called isoperimetric. In this paper the author studies the classical isoperimetric problem in a cylinder of type \({\mathbb R}\times {S}^n\) and proves the following properties for an isoperimetric region: (1) it is regular (has analytic boundary) and connected; (2) its intersection with the slices \(\{x\}\times {S}^n\) are empty, points, the whole slice, or geodesic balls of \(\{x\}\times {S}^n;\) (3) it is \(\text{ O}(n)\) invariant; and (4) it is either a cylindrical section (\(\Omega=[a,b]\times {S}^n\)) or congruent to a ball-type region. The author provides a more precise description of an isoperimetric region for \(n=2\).