On The Arithmetic-Geometric Mean Inequality And Its Relationship To Linear Programming, Matrix Scaling, And Gordan's Theorem

. 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 al..