  We present a density-functional (DF) study of structures of Si cage clusters
that encapsulate metal atoms. As a prototypical example, the case of WSi$_{n}$
clusters is shown. To obtain the low-energy clusters in efficient and unbiased
ways, genetic-like geometry updates have been performed for generating inputs
for subsequent local optimizations within DF calculations. Well-defined cages
occur for a certain range of $n$. Such cages are modelled as simple 3-polytopes
where the numbers of their inner diagonals close to the metal atom are
maximized.
