http://openproblemgarden.org/op/fat_4_polytopes Comments for "Fat 4-polytopes" en http://openproblemgarden.org/op/fat_4_polytopes <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/eppstein_david">Eppstein</a>; <a href="/category/kuperberg_greg">Kuperberg</a>; <a href="/category/ziegler_gunter_m">Ziegler</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/geometry">Geometry</a> » <a href="/category/polytopes">Polytopes</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>The <em>fatness</em> of a 4-<a href="http://en.wikipedia.org/wiki/polytope">polytope</a> <img class="teximage" src="/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png" alt="$ P $" /> is defined to be <img class="teximage" src="/files/tex/520b0af3282aea53dbec668935163e110d9e5178.png" alt="$ (f_1 + f_2)/(f_0 + f_3) $" /> where <img class="teximage" src="/files/tex/3d8d4274a2a88f65b70123aab5f541cbfb114eb2.png" alt="$ f_i $" /> is the number of faces of <img class="teximage" src="/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png" alt="$ P $" /> of dimension <img class="teximage" src="/files/tex/ca49c241ece07915c97a31774a977841c6f0414c.png" alt="$ i $" />.</p> <div class="envtheorem"><b>Question</b>&nbsp;&nbsp; Does there exist a fixed constant <img class="teximage" src="/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png" alt="$ c $" /> so that every convex 4-polytope has fatness at most <img class="teximage" src="/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png" alt="$ c $" />? </div> </tr></td> </table> </td> </tr> </table> Eppstein, David Kuperberg, Greg Ziegler, Gunter M. f-vector polytope Geometry Polytopes http://openproblemgarden.org/op/fat_4_polytopes#comment Sat, 12 May 2007 10:34:09 +0200 mdevos 316 at http://openproblemgarden.org