Extremal elements of a sublattice of the majorization lattice and approximate majorization
- Massri, Cesar Dario
- Bellomo, Guido
- Holik, Federico Hernán
- Bosyk, Gustavo Martin
DOIAbstract
Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball B p ǫ (x) with p > 1 formed by the probability vectors whose ℓ p -norm distance to the center x is less than or equal to a radius ǫ. Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. Unlike the case p = 1 previously discussed in the literature, we nd that the extremal probability vectors, in general, do not exist for the closed balls B p ǫ (x) with 1 < p < ∞. On the other hand, we show that B ∞ ǫ (x) is a complete sublattice of the majorization lattice. As a consequence, this ball also has extremal elements. In addition, we give an explicit characterization of those extremal...
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