Polynomial optimization problems are problems of optimizing a multivariate polynomial over the feasible set defined by a finite number of polynomial inequalities. It encompasses many problems within various fields of mathematics, e.g., binary optimization, mixed-integer linear programming, global optimization and partial differential inequalities. Problems of polynomial optimization can be equivalently reformulated as problems over the convex cone of non-negative polynomials. In this paper, the geometric and topological properties of a cone of polynomials non-negative on a given set and the respective dual cone are studied.