  We study optimal simple second-order cone representations (a particular
subclass of second-order cone representations) for weighted geometric means,
which turns out to be closely related to minimum mediated sets. Several lower
and upper bounds on the size of optimal simple 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 simple 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 simple
second-order cone representation. Finally, applications to polynomial
optimization, matrix optimization, and quantum information are provided.
