We study optimal second-order cone representations for weighted geometric means, which turns out to be closely related to minimum mediated sets. Several lower bounds and upper bounds on the size of optimal second-order cone representations are proved. In the case of bivariate weighted geometric means (equivalently, one dimensional mediated sets), we are able to prove the exact size of an optimal second-order cone representation and give an algorithm to compute one. In the genenal case, fast heuristic algorithms and traversal algorithms are proposed to compute an approximately optimal second-order cone representation. Finally, applications to polynomial optimization, matrix optimization and quantum information are provided.
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