Nearly simple polytopes

Taking as our domain the class of all polytopes with a given dimension and a given number of vertices, we begin with the optimisation questions: what is the minimum possible number of edges? What about higher dimensional faces? (The corresponding questions about maxima are easy.) Moreover, is the range of values a complete interval, or are there gaps? This study leads to a new invariant for polytopes, the excess degree, and to some surprising structural results about polytopes which are nearly simple.