06-21

Instability of discrete point sets

by Achill Schürmann

 

Preprint series: 06-21, Preprints

MSC:
52C25 Rigidity and flexibility of structures, See also {70B15}

 

Abstract: Let $X$ be a discrete subset of Euclidean $d$-space. We allow subsequently continuous movements of single elements, whenever the minimum distance to other elements does not decrease. We discuss the question, if it is possible to move all elements of $X$ in this way, for example after removing a finite subset $Y$ from $X$. Although it is not possible in general, we show the existence of such finite subsets $Y$ for many discrete sets $X$, including all lattices. We define the \textit {instability degree} of $X$ as the minimum cardinality of such a subset $Y$ and show that the maximum instability degree among lattices is attained by perfect lattices. Moreover, we discuss the $3$-dimensional case in detail.

Keywords: instability, discrete point sets


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