Find the intersection of two or more polygons in terms of linear inequalities

Question

Given two or more closed polygons each defined by a system of linear inequalities, is there any method by which their intersection polygon may be determined, also in terms of a system of linear inequalities? It is certain that all the inequalities in the intersection set will also be members of at least one of the input sets. I'm interested in a method that could be represented as the solution of single matrix equation, rather than a stepwise process.

Answer 1

In general, the number of inequalities used to describe one convex polytope (high dimensional polygon) may not equal the number needed to describe another, even if they share the same dimension.

Consider a triangle, which in $\mathbb{R}^2$ requires 3, while a pentagon requires 5.

If $A$ is the set of linear inequalities for one polygon and $B$ is the set of linear inequalities for another, you could consider $A \cup B$ to describe their intersection.

Eliminating redundancies in inequalities can be done via linear programming.