Lutwak, Yang and Zhang defined the cone volume functional U over convex polytopes in R^n containing the origin in their interiors, and conjectured that the greatest lower bound on the ratio of this centro-affine invariant U to volume V is attained by parallelotopes. In this paper, we give affirmative answers to the conjecture in R^2 and R^3. Some new sharp inequalities characterizing parallelotopes in Rn are established. Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes in Rn is provided.

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We extend recent results by Pisier on$K$-subcouples, i.e. subcouples of an interpolation couple that preserve the$K$-functional (up to constants) and corresponding notions for quotient couples. Examples include interpolation (in the pointwise sense) and a reinterpretation of the Adamyan-Arov-Krein theorem for Hankel operators.

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The approach combines second and fourth order statistics to perform BSS of instantaneous mixtures. It applies for any number of receivers if they are as many as sources. It is a batch algorithm that uses non-Gaussianity and stationarity of source signals. It is linear algebra based direct method, reliable and robust, though large dimensions of sources may slow down the computation significantly. It is however limited to instantaneous mixtures.