Let S be a sphere of unit radius in three dimensional Euclidean space, R^3. Given a positive real number e, does there always exist a convex polyhedron P in R^3 such that: (1) S is a subset of P (2) The boundary of P is homeomorphic to the boundary of S (3) The volume of P does not exceed the volume of S by more than e? It is not required that S be tangent to any of the faces of P.
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For small $\epsilon$ let $Q$ be the convex symmetric hull of a finite $\epsilon$ net for the boundary of $S$ and let $P=(1+\epsilon) Q$. 


This is proposition 17 of book 12 of Euclid's elements. 

