Polyhedral theory is a field of mathematics and geometry dedicated to the study of polyhedra and their higher-dimensional analogues, focusing on their geometric, combinatorial, and algebraic properties, particularly within the context of convex sets, linear inequalities, and their applications in optimization and discrete mathematics. It investigates the structure of convex polyhedra as intersections of half-spaces or convex hulls of finite point sets, establishing fundamental theorems regarding their facets, vertices, and duality, which are crucial for understanding and solving linear and integer programming problems.