  We consider symmetry operations on the four-dimensional vector space that is
spanned by the local versions of the Minkowski functionals (or fundamental
measures): volume, surface, integral mean curvature, and Euler characteristic,
of an underlying three-dimensional geometry. A bilinear combination of the
measures is used as a (pseudo) metric with ++-- signature, represented by a 4x4
matrix with unit entries on the counter diagonal. Six different types of linear
automorphisms are shown to leave the metric invariant. Their generators form a
Lie algebra that can be grouped into two mutually commuting triples with
non-trivial structure constants. We supplement these six isometric operations
by further ten transformations that have a metamorphic (altering) effect on the
underlying geometry. When grouped together, four different linear combinations
of the metamorphic generators form a previously obtained third-rank tensor.
This is shown to describe four different types of mutually commuting "shifting"
operations in fundamental measure space. The relevance for fundamental measures
density functional theory is discussed briefly.
