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We show that a partition of the unity (or POVM) on a Hilbert space that is almost orthogonal is close to an orthogonal POVM in the same von Neumann algebra. This generalizes to infinite dimension previous results in matrix algebras by Kempe–Vidick and Ji–Natarajan–Vidick–Wright–Yuen. Quantitatively, our result are also finer, as we obtain a linear dependance, which is optimal. We also generalize to infinite dimension a duality result between POVMs and minimal majorants of finite subsets in the predual of a von Neumann algebra. [ABSTRACT FROM AUTHOR]

We present a new proof of Thurston's theorem that the unit ball of a seminorm on R d taking integer values on Z d is a polyhedron defined by finitely many inequalities with integer coefficients. [ABSTRACT FROM AUTHOR]