by Achill Schürmann.

**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