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Abstract

Given any real numbers $1 \lt p \lt q$, we study the norm ratio (i.e. the ratio between the $q$-norm and the $p$-norm) of marginals of centered convex bodies. We first show that some marginal of the simplex maximizes the said ratio in the class of $n$-dimensional centered convex bodies. We then pass to the dimension-independent (i.e. log-concave) case where we find a 1-parameter family of random variables in which the maximum ratio must be attained, and find the exact maximizer of the ratio when $p=2$ and $q$ is even. In addition, we find another interesting maximization property of marginals of the simplex involving functions with positive third derivatives.