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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
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1 Answer

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.

However, deltahedra are simplicial polyhedra whose faces are all equilateral triangles, so maybe you are asking whether the simplicial polytopes dual to associahedra may be realized with faces that are regular simplices? Then the results of this paper 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 the right number 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.

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.

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Thanks, you confirm that dual polytopes exist for associahedra in all dimensions and that minimal surfaces/soap films (nice paper) don't provide an analog to the Thomson problem in higher dimensions that can be related to the duals. I'd be happy learning that the duals with vertices occupied by identical ions provide a local energy minimum for a Thomson-like problem in 4-D and beyond (or even more generally, solutions to some min/max problem that I can then relate back to the associahedra somehow). The (stable) equilibria are those achieved after dropping ions on the sphere per original MOQ. –  Tom Copeland Mar 15 '12 at 0:17
I see, what you're asking is whether duals of associahedra may provide equilibrium positions for such problems, not whether they are deltahedra or not. Could you not check this at least in 4D with the data that Henry Cohn linked to in the comments aimath.org/data/paper/BBCGKS2006 ? –  j.c. Mar 15 '12 at 0:46
For example, the dual of the 4D associahedron has 14 vertices, so you'd have to check whether the coordinates here aimath.org/data/paper/BBCGKS2006/4-14.txt could be interpreted as the vertices of such a polytope. What I envision is using something like a (3-spherical) Delaunay triangulation to define a polytope; then one can check whether the combinatorial structure happens to be that of the dual of the 4D associahedron. Regardless of whether this particular case happens to work out, I think you need some Delaunay-like construction to get polytopes from equilibrium positions of points. –  j.c. Mar 15 '12 at 2:54
I'm just throwing suggestions out there. I don't think I'll follow up on them personally. –  j.c. Mar 15 '12 at 14:18
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