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It is well known that a sphere has the lowest surface to volume ratio. However, a related question is: What is the shape that gives the lowest surface to volume ratio if you do not include the top in the surface. That is, what is the maximal volume of an uncovered vessel of a fixed surface area?

To be more specific with the definition of top or cover -- I regard this as a continuous manifold which allow holes. The holes do not contribute to the surface, but the volume I consider is only the one up to (up defined by the direction of gravity) the first hole. The answer thus depends on the direction of the vessel. An upside-down water glass would have an infinite ratio.

For example, a cylinder whose height is equal to its radius and a missing top basis, has a volume $\pi R^2 h = \pi R^3$ and a surface (without cover) of $\pi R^2 + 2\pi R h = 3 \pi R^2$. If we fix the volume to unity, the surface in this case is $3 \sqrt[3]{\pi}$.

In comparison, a half sphere of unity volume has a smaller surface, i.e. $\sqrt[3]{18\pi}$ .

However, it's clear this is not the best shape -- a sphere cut just a little higher the equator beats the half sphere.

So, what is the shape that gives the lowest surface to volume ratio if you do not include the top in the surface?


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I'm curious-- will punching an infinetisimally small hole on a spherical vessel do the trick? – Frenzy Li Jul 2 '12 at 12:43
"A sphere cut a big above half its volume beats the half sphere." True. Now take that to its logical conclusion: A sphere cut arbitrarily-close to its North Pole gets arbitrarily-close to the optimal ratio. On the other hand, if you say that the hole must be non-infinitesimal in some way --say, a circle of a given radius-- then one can construct an arbitrarily-large sphere with such a hole, ever-improving the surface-to-volume ratio. – Blue Jul 2 '12 at 12:43
No. If we limit ourselves to spherical caps, the optimal solution is not the full sphere with an infinitesimal hole, but rather a cap cut at height which is 3/2 times the radius. – Eli Jul 2 '12 at 14:01
sorry, my mistake. Optimizing with constant volume gives the half sphere as the optimal spherical cap. Still, I wonder what's the best shape in general. – Eli Jul 2 '12 at 14:27

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