Isovolume Problem

Find the surface enclosing the maximum volume per unit surface area, . The solution is a sphere, which has

The fact that a sphere solves the isovolume problem was only proved as recently as 1882 by Schwarz (Haas 2000).

REFERENCES:

Bogomolny, A. "Isoperimetric Theorem and Inequality." http://www.cut-the-knot.org/do_you_know/isoperimetric.shtml.

Haas, J. "General Double Bubble Conjecture in  Solved." Focus: The Newsletter of the Math. Assoc. Amer., No. 5, pp. 4-5, May/June 2000.

Isenberg, C. "The Maximum Volume Contained by a Closed Surface of Fixed Area." Appendix VI in The Science of Soap Films and Soap Bubbles. New York: Dover, pp. 174-177, 1992.

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 214, 1999.