Cones, dual cones and polytopes

In this first block the polytope and cone algebraic structures are presented, showing some illustrative examples that facilitate their understanding. The concept of a dual cone of a given cone is also presented. As a fundamental element for the development of the course, the algorithm for obtaining the dual cone of a given cone is described, which although designed to obtain the dual cone, will serve as the basis for solving all the other problems that arise in the course. It is shown that the vector space is a particular case of cone and that it is enough to add one more generator to the base in order to express the vectors of the vector space as cone, that is, generated by nonnegative linear combinations. Finally, the standard form of a cone is defined as the sum of its components of vector space and acute cone, which allows expressing the cone in its minimal form.