Locally translation-isometric and multi-isometric sets as generalizations of objective structures

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Motivated by a simplified model for atomic structures, and based on the concept of objective structures in R3, we introduce a type of subsets of Rn that we call locally translation-isometric sets (LTI sets). These sets are defined by the property that the ɛ-neighborhoods around every point in them are isometric copies of each other. For n=3, in the case ɛ=∞ they are simply objective structures and correspond (for n∈N) to the orbits of symmetry groups with translations. We provide a partial characterization of LTI sets with finite ɛ, and discuss some interesting examples and results. We generalize the concept further, in particular to locally multi-isometric sets, i.e. sets where the ɛ-neighborhoods of every point look like one of finitely many sets. We show that these generalizations of objective structures can describe even exotic atomic configurations quite well.