Linear programming, best way of solving discrete Optimization problem.. either exactly (when the solution can be found polynomially), or approximately, when NP-hard. There is a conjecture that is unproved, to show this.
Information Theory and Polyhedral Combinatorics
http://sci-hub.cc/10.1109/allerton.2015.7447134. Extension theory
In Figure 1 we provide a simple example showing that higher dimensional representations may require fewer inequalities