Quadratic forms positive on the cone and quadratic duality

A general pattern is outlined for the study of quadratic forms which are positive on a cone in a finite-dimensional linear space. Quadratic duality theorems are proved. An example related to the $S$-procedure in controle theory and to the Pareto-optimum is thoroughly considered. Geometric proof of the Hausdorff–Toeplitz convexity theorem is discussed.