2011/12/20 by JimÉnez, C. Hugo, NaszÓdi, MÁrton, Villa, Rafael · 1 citation
#46B06 #46B09 #46b07 #52A20 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1112.4765
In this note we study how a concentration phenomenon can be transmitted from one measure μ to a push-forward measure ν. In the first part, we push forward μ by π:supp(μ)→ \Ren, where πx=\fracx\normxL\normxK, and obtain a concentration inequality in terms of the medians of the given norms (with respect to μ) and the Banach-Mazur distance between them. This approach is finer than simply bounding the concentration of the push forward measure in terms of the Banach-Mazur distance between K and L. As a corollary we show that any normed probability space with good concentration is far from any high dimensional subspace of the cube. In the second part, two measures μ and ν are given, both related to the norm \norm⋅L, obtaining a concentration inequality in which it is involved the Banach-Mazur distance between K and L and the Lipschitz constant of the map that pushes forward μ into ν. As an application, we obtain a concentration inequality for the cross polytope with respect to the normalized Lebesgue measure and the ℓ1 norm.