Sum of Square Proof for Brascamp-Lieb Type Inequality

Brascamp-Lieb inequalities [5] is an important mathematical tool in analysis, geometry and information theory. There are various ways to prove Brascamp-Lieb inequality such as heat flow method [4], Brownian motion [11] and subadditivity of the entropy [6]. While Brascamp-Lieb inequality is originally stated in Euclidean Space, [8] discussed Brascamp-Lieb inequality for discrete Abelian group and [3] discussed Brascamp-Lieb inequality for Markov semigroups. Many mathematical inequalities can be formulated as algebraic inequalities which asserts some given polynomial is nonnegative. In 1927, Artin proved that any nonnegative polynomial can be represented as a sum of squares of rational functions [10], which can be further formulated as a polynomial certificate of the nonnegativity of the polynomial. This is a Sum-of-Square proof of the inequality. The Sum-of-Square proof can be captured by Sum-of-Square algorithm which is a powerful tool for optimization and computer aided proof. For more about Sum-of-Square algorithm, see [2]. In this paper, we give a Sum-of-Square proof for some special settings of BrascampLieb inequality following [9] and [4] and discuss some applications of Brascamp-Lieb inequality on Abelian group and Euclidean Sphere. If the original description of the inequality has constant degree and d is constant, the degree of the proof is also constant. Therefore, low degree sum of square algorithm can fully capture the power of low degree finite Brascamp-Lieb inequality.