  We introduce a natural notion of depth that applies to individual cutting
planes as well as entire families. This depth has nice properties that make it
easy to work with theoretically, and we argue that it is a good proxy for the
practical strength of cutting planes. In particular, we show that its value
lies in a well-defined interval, and we give parametric upper bounds on the
depth of two prominent types of cutting planes: split cuts and intersection
cuts from a simplex tableau. Interestingly, these parametric bounds can help
explain behaviors that have been observed computationally. For polyhedra, the
depth of an individual cutting plane can be computed in polynomial time through
an LP formulation, and we show that it can be computed in closed form in the
case of corner polyhedra.
