### 2006-21

#### Instability of discrete point sets

Series: 2006-21, Preprints

MSC:
52C25 Rigidity and flexibility of structures

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