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Isovolume Problem

The isovolume problem asks one to find the surface enclosing the maximum volume per unit surface area, I=V/S. The solution is a sphere, which has

I_(sphere)=(4/3pir^3)/(4pir^2)=1/3r.

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

See also

Dido's Problem, Double Bubble, Isoperimetric Problem, Sphere, Surface Area, Volume

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References

Bogomolny, A. "Isoperimetric Theorem and Inequality." https://www.cut-the-knot.org/do_you_know/isoperimetric.shtml.Haas, J. "General Double Bubble Conjecture in R^3 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.

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Isovolume Problem

Cite this as:

Weisstein, Eric W. "Isovolume Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IsovolumeProblem.html

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