Minimizing blossoms under symmetric linear constraints

In this paper, we show that there exists a close dependence between the control polygon of a polynomial and the minimum of its blossom under symmetric linear constraints. We consider a given minimization problem P, for which a unique solution will be a point δ on the Bézier curve. For the minimization function f, two sufficient conditions exist that ensure the uniqueness of the solution, namely, the concavity of the control polygon of the polynomial and the characteristics of the Polya frequency-control polygon where the minimum coincides with a critical point of the polynomial. The use of the blossoming theory provides us with a useful geometrical interpretation of the minimization problem. In addition, this minimization approach leads us to a new method of discovering inequalities about the elementary symmetric polynomials.