  A geometric understanding of entanglement is proposed based on local
measurements. Taking recourse to the general structure of density matrices in
the framework of Euclidean geometry, we first illustrate our approach for
bipartite Werner states. It is demonstrated that separable states satisfy
certain geometric constraints that entangled states do not. A separability
criterion for multiparty Werner states of arbitrary dimension is derived. This
approach can be used to determine separability across any bipartition of a
general density matrix and leads naturally to a computable measure of
entanglement for multiparty pure states. It is known that all density matrices
within a certain distance of the normalized identity are separable. This
distance is determined for a general setting of $n$ qudits, each of dimension
$d$.
