Abstract: Many industrial problems fit the model of linear programming. The design and analysis of linear programming algorithms lead to the study of polyhedra, namely generalizations of three dimensional solids such as cubes and pyramids. The talk focuses on a particular geometric question about polyhedra, mostly staying in three dimensions, where the polyhedra can be visualized. The goal is to demonstrate how mathematical research proceeds through raising questions, making conjectures, developing constructions, and proving results. No previous knowledge of linear programming or polyhedra is assumed.