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3. Delone Sets

A Delone set is a subset of $\mathbb{R}^n$ satisfying the following properties: There exists a pair of positive constants $(r,R)$ such that every ball of radius r contains at most one point of $X$ and ever ball of radius $R$ contains at least one point of $X$.
    1. Building skeletal complexes

          The notion of a skeletal complex can be naturally extended to Delone sets.

      Problem 3.1.

      [Schulte] Study skeletal complexes based on Delone sets.
        • Problem 3.2.

          [Dolbilin] For $d \ge 4$, given a Delone set $(R,r)$ for which all $2R$- clusters are pairwise congruent, find upper bound on the order of the symmetry group of the clusters which does not depend on $R/r$.
            •     In 3 dimensions, it is known that if all $10R$ clusters are equivalent, the Delone set is regular.

              Problem 3.3.

              [Dolbilin] Can this result be improved for $\rho < 10R$, e.g. $\rho = 4R$? Is it possible to show similar results for full dimensional $r$-sets with conditions weaker than the $R$ condition?
                  Example presented on Wed 9/21/2016 giving a construction of non-regular sets
                • Cycles in skeletal complexes

                  Problem 3.4.

                  [Dolbilin] What are the applications and physical significance of the cycles in skeletal polygonal complexes?

                      Cite this as: AimPL: Soft packings, nested clusters, and condensed matter, available at http://aimpl.org/softpack.