Max/min problems related to associahedra or their duals (ions on balls revisited) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:05:13Z http://mathoverflow.net/feeds/question/91168 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91168/max-min-problems-related-to-associahedra-or-their-duals-ions-on-balls-revisited Max/min problems related to associahedra or their duals (ions on balls revisited) Tom Copeland 2012-03-14T12:22:32Z 2012-03-15T01:46:49Z <p>Original motivation: This is a follow-up question to and generalization of MO <a href="http://mathoverflow.net/questions/78877/equilibrium-configurations-of-ions-on-n-dim-balls" rel="nofollow">Q78877</a> on equilibrium configurations of ions on n-Dim balls. Henry Cohn gave an excellent answer dispelling my naive intuition/hope that 14 ions would configure into the vertices of a Stasheff <a href="http://en.wikipedia.org/wiki/Associahedron" rel="nofollow">associahedron</a> on a 3-Dim ball under the influence of a Coulomb potential (<a href="http://en.wikipedia.org/wiki/Thomson_problem" rel="nofollow">Thomson problem</a>). As he remarks, the ions would configure into the vertices of <a href="http://en.wikipedia.org/wiki/Deltahedra" rel="nofollow">deltahedra</a> (with simplicial/triangular facets) on the 3-D ball. However, on a web page by Maurice Starck, I just noticed that a convex deltahedron with 9 vertices has 21 edges and 14 faces-the dual polyhedron to the 3-D associahedron! The 2-D case, the self-dual pentagon, is analogous. Is there a 4-D analog, i.e., does the <a href="http://oeis.org/A033282" rel="nofollow">4-D dual polytope</a> with 14 ions at its vertices satisfy a 4-D Thompson-like problem?</p> <p>Prompted by JC's reply, I'd really like to know more generally of any (natural/enlightening) max/min problems with solutions involving the associahedra or their dual polytopes. </p> http://mathoverflow.net/questions/91168/max-min-problems-related-to-associahedra-or-their-duals-ions-on-balls-revisited/91223#91223 Answer by jc for Max/min problems related to associahedra or their duals (ions on balls revisited) jc 2012-03-14T21:31:11Z 2012-03-14T21:42:15Z <p>If your question is simply whether the 4-D associahedron is dual to a <a href="http://en.wikipedia.org/wiki/Simplicial_polytope" rel="nofollow">simplicial polytope</a>, the answer is yes, because all associahedra are <a href="http://en.wikipedia.org/wiki/Simple_polytope" rel="nofollow">simple polytopes</a>. To see this, note that the vertices of \$K_{d+2}\$ correspond to strings of \$d+2\$ letters "saturated" by \$d\$ pairs of parentheses. The \$d\$ edges in the star of a vertex therefore correspond to removing any one of those \$d\$ pairs of parentheses.</p> <p>However, deltahedra are simplicial polyhedra whose faces are all <em>equilateral</em> triangles, so maybe you are asking whether the simplicial polytopes dual to associahedra may be realized with faces that are <em>regular</em> simplices? Then the results of <a href="http://www.emis.de/mirror/IMU/Logo/tmp/torus.math.uiuc.edu/jms/Papers/delta.pdf" rel="nofollow">this paper</a> of John Sullivan's which classifies "convex deltatopes" imply that the duals of higher dimensional associahedra cannot be convex deltatopes (I checked that the convex deltatopes he constructs do not have <a href="http://en.wikipedia.org/wiki/Catalan_number" rel="nofollow">the right number</a> of faces once the dimension is greater than 3), and I suspect that one may be able to show that the dual simplicial polytopes of associahedra can't be made into deltatopes at all.</p> <p>On a side note I recommend changing the title of the question and making it more clear in the body precisely what you are asking. The reference to equilibrium positions of ions, while interesting, threw me off.</p>