# Max/min problems related to associahedra or their duals (ions on balls revisited)

Original motivation: This is a follow-up question to and generalization of MO Q78877 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 associahedron on a 3-Dim ball under the influence of a Coulomb potential (Thomson problem). As he remarks, the ions would configure into the vertices of deltahedra (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 4-D dual polytope with 14 ions at its vertices satisfy a 4-D Thompson-like problem?

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.

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Harry Cohn of Columbia Pictures? –  Igor Rivin Mar 14 '12 at 15:03
The dual for the 3-D associahedron is depicted in en.m.wikipedia.org/wiki/Triaugmented_triangular_prism. –  Tom Copeland Jan 10 at 3:01

If your question is simply whether the 4-D associahedron is dual to a simplicial polytope, the answer is yes, because all associahedra are simple polytopes. 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.