Pii: S0167-8396(02)00127-9

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.  2002 Published by Elsevier Science B.V.