Polytope shaping while preserving invariance

We present a framework on how to dynamically change the shape of invariant polytopes via elementary operations, namely, by adding a vertex or a linear inequality in their description, and at the same time maintain invariance. We work on the face lattice of the polytope, and introduce the cone lattice which embeds in an efficient way the geometric and combinatorial structure of the polytope. Combining the above, we can characterize a priori the complexity of the set resulting from either elementary operation. Importantly, using order-theoretic arguments, we identify the parts of the face and cone lattice that must be recalculated in every operation. Finally, we recall and extend equivalent algebraic conditions for preserving invariance of polytopes for a few families of dynamical systems.