
## 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.