  In this paper, we establish a connection between zonoids (a concept from
classical convex geometry) and the distinguishability norms associated to
quantum measurements, or POVMs (Positive Operator-Valued Measures), recently
introduced in quantum information theory.
  This correspondence allows us to state and prove the POVM version of
classical results from the local theory of Banach spaces about the
approximation of zonoids by zonotopes. We show that on $\mathbf{C}^d$, the
uniform POVM (the most symmetric POVM) can be sparsified, i.e. approximated by
a discrete POVM, the latter having only $O(d^2)$ outcomes. We also show that
similar (but weaker) approximation results actually hold for any POVM on
$\mathbf{C}^d$.
  By defining an appropriate notion of tensor product for zonoids, we are then
able to extend our results to the multipartite setting: we show, roughly
speaking, that local POVMs may be sparsified locally. In particular, the local
uniform POVM on $\mathbf{C}^{d_1}\otimes\cdots\otimes\mathbf{C}^{d_k}$ can be
approximated by a discrete POVM which is local and has
$O(d_1^2\times\cdots\times d_k^2)$ outcomes.
