Push forward measures and concentration phenomena

In this note we study how a concentration phenomenon can be transferred from one measure μ to a push‐forward measure ν. In the first part, we push forward μ by , where , 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 consequence we show that any normed probability space with exponential type concentration is far (even in an average sense) from subspaces of . The sharpness of this result is shown by considering the spaces.