A question about convex polyhedra - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T07:48:35Z http://mathoverflow.net/feeds/question/49052 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49052/a-question-about-convex-polyhedra A question about convex polyhedra Garabed Gulbenkian 2010-12-11T15:58:11Z 2010-12-11T16:39:31Z <p>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.</p> http://mathoverflow.net/questions/49052/a-question-about-convex-polyhedra/49053#49053 Answer by Bill Johnson for A question about convex polyhedra Bill Johnson 2010-12-11T16:19:12Z 2010-12-11T16:19:12Z <p>For small $\epsilon$ let $Q$ be the convex symmetric hull of a finite $\epsilon$ net for the boundary of $S$ and let $P=(1+\epsilon) Q$. </p> http://mathoverflow.net/questions/49052/a-question-about-convex-polyhedra/49055#49055 Answer by Richard Borcherds for A question about convex polyhedra Richard Borcherds 2010-12-11T16:39:31Z 2010-12-11T16:39:31Z <p>This is proposition 17 of book 12 of Euclid's elements. </p>