  Certain many-particle Hardy inequalities are derived in a simple and
systematic way using the so-called ground state representation for the
Laplacian on a subdomain of $\mathbb{R}^n$. This includes geometric extensions
of the standard Hardy inequalities to involve volumes of simplices spanned by a
subset of points. Clifford/multilinear algebra is employed to simplify
geometric computations. These results and the techniques involved are relevant
for classes of exactly solvable quantum systems such as the Calogero-Sutherland
models and their higher-dimensional generalizations, as well as for membrane
matrix models, and models of more complicated particle interactions of
geometric character.
