We obtain an effective lower bound on the distance of the sum of co-adjoint orbits from the origin. Even when the distance is zero (thus the symplectic quotient is well defined) our result gives a nontrivial constraint on these co-adjoint orbits. In the particular case of unitary groups, we obtain the quadratic inequality for eigenvalues of Hermitian matrices satisfying A + B = C. This quadratic inequality can be interpreted as the Chern number inequality for semi-stable reflexive toric sheaves.

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We prove a compactness principle for the anisotropic formulation of the Plateau’s problem in codimension one, along the same lines of previous works of the authors [DGM14, DPDRG15]. In particular, we perform a new strategy for proving the rectifiability of the minimal set, avoiding the Preiss’ Rectifiability Theorem.