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| Math Central | Quandaries & Queries |
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Question from Dutch: I'm searching for the proof that the sphere has the smallest volume of any figure and maximizes the volume of any figure. thanks |
Dutch,
The problem you refer to is the 3-dimensional version of the isoperimetric inequality. Specifically, the problem is to show that a sphere is the surface that encloses the maximum volume per unit surface area. There are many elementary treatments of the problem in the plane (to show that the curve of unit length that encloses the largest possible area is the circle). MATH WORLD
http://mathworld.wolfram.com/IsovolumeProblem.html
calls your problem "the Isovolume Problem" and provides several references, but no proof. More references are provided in the Wikipedia,
http://en.wikipedia.org/wiki/Isoperimetry.
Chris
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