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).
Isovolume Problem
See also
Dido's Problem, Double Bubble, Isoperimetric Problem, Sphere, Surface Area, VolumeExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Isovolume ProblemCite this as:
Weisstein, Eric W. "Isovolume Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IsovolumeProblem.html