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We describe the behavior of certain strictly plurisubharmonic functions near some real hypersurfaces in ℂ^{$n$},$n$≥3. Given a hypersurface we study continuous plurisubharmonic functions which are zero on the hypersurface and have Monge–Ampère mass greater than one in a one-sided neighborhood of the hypersurface. If we can find complex curves which have sufficiently high contact order with the hypersurface then the plurisubharmonic functions we study cannot be globally Lipschitz in the one-sided neighborhood.

We show almost sure simplicity of eigenvalues for several models of Anderson-type random Schrödinger operators, extending methods introduced by Simon for the discrete Anderson model. These methods work throughout the spectrum and are not restricted to the localization regime. We establish general criteria for the simplicity of eigenvalues which can be interpreted as separately excluding the absence of local and global symmetries, respectively. The criteria are applied to Anderson models with matrix-valued potential as well as with single-site potentials supported on a finite box.

By using the method of mixed volumes, we give sharp bounds for inclusion measures of convex bodies in n-dimensional Euclidean space. In the special cases where the random convex body is the unit ball or when n = 3, neater and simpler bounds are obtained. All the associated inequalities proved are new isoperimetrictype inequalities.

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