# A question about convex polyhedra

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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(2) S has no boundary, so I assume you mean boundary of P is homeomorphic to S. Otherwise, the answer is YES, and this looks an awful lot like a homework problem. –  Igor Rivin Dec 11 '10 at 16:11
I interpreted "sphere" to mean "ball". I would not have answered except the OP gives his name and has asked some reasonable questions in the past. –  Bill Johnson Dec 11 '10 at 16:21

For small $\epsilon$ let $Q$ be the convex symmetric hull of a finite $\epsilon$ net for the boundary of $S$ and let $P=(1+\epsilon) Q$.