Abstract
It is a classical inequality that the minimum of the ratio of the (weighted) arithmetic mean to the geometric mean of a set of positive variables is equal to one, and is attained at the center of the positivity cone. While there are numerous proofs of this fundamental homogeneous inequality, in the presence of an arbitrary subspace, and/or the replacement of the arithmetic mean with an arbitrary linear form, the new minimization is a nontrivial problem. We prove a generalization of this inequality, also relating it to linear programming, to the diagonal matrix scaling problem, as well as to Gordan's theorem. Linear programming is equivalent to the search for a nontrivial zero of a linear or positive semidefinite quadratic form over the nonnegative points of a given subspace. The goal of this paper is to present these intricate, surprising, and significant relationships, called scaling dualities, and via an elementary proof. Also, to introduce two conceptually simple polynomialtime algorithms that are based on the scaling dualities, significant bounds, as well as Nesterov and Nemirovskii's machinery of self-concordance. The algorithms are simultaneously applicable to linear programming, to the computation of a separating hyperplane, to the diagonal matrix scaling problem, and to the minimization of the arithmetic-geometric mean ratio over the positive points of an arbitrary subspace. The scaling dualities, the bounds, and the algorithms are special cases of a much more general theory on convex programming, developed by the author. For instance, via the scaling dualities semidefinite programming is a problem dual to a generalization of the classical trace-determinant ratio minimization over the positive definite points of a given subspace of the Hilbert space of symmetric matrices.