Smallest containing simplex - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T07:39:34Z http://mathoverflow.net/feeds/question/82331 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82331/smallest-containing-simplex Smallest containing simplex Vladimir Reshetnikov 2011-12-01T00:02:36Z 2013-05-22T20:33:57Z <p>Let \$V_n\$ be the least real number such that for every convex subset of \$\mathbb{R}^n\$ with hypervolume \$1\$ there is a containing simplex with hypervolume \$V_n\$. What is known about \$V_n\$? Is there a known general formula? If not, then what is the best known bounds on \$V_n\$?</p> http://mathoverflow.net/questions/82331/smallest-containing-simplex/82356#82356 Answer by Michael Biro for Smallest containing simplex Michael Biro 2011-12-01T08:33:56Z 2011-12-01T08:40:32Z <p>The paper <a href="http://www.springerlink.com/content/cu7rvr8m7vnlhgtr/" rel="nofollow">Parallelotopes of Maximum Volume in a Simplex</a> by Lassak gives the maximum possible volume of a parallelotope in a simplex as \$n!/n^n\$ times the volume of the simplex. This gives us a bound of \$V_n \geq n^n/n!\$, which I suspect is tight.</p> http://mathoverflow.net/questions/82331/smallest-containing-simplex/131505#131505 Answer by TauMu for Smallest containing simplex TauMu 2013-05-22T20:33:57Z 2013-05-22T20:33:57Z <p>The problem seems to be still open even for \$n=3\$:<br> <em>Weisstein, Eric W. <a href="http://mathworld.wolfram.com/TetrahedronCircumscribing.html" rel="nofollow">"Tetrahedron Circumscribing."</a></em></p>