What's So Special about Minkowski Addition?

One of the most widely applied operations in mathematics is vector addition. It can be used to define an operation between sets: Given two sets K and L, one defines K+L to be the set of all vector sums x+y, where x is in K and y is in L. As an operation between sets, vector addition is usually called Minkowski addition, and is the single most important set operation in geometry. There are several other well-known and useful operations between sets in geometry - for example, L_p addition, radial addition, and Blaschke addition - but surprisingly few. Why is this? What is so special about the known operations, Minkowski addition in particular?
The talk is a report on a long-term project, still in progress, with Daniel Hug and Wolfgang Weil of the University of Karlsruhe. The general goal is to achieve a proper understanding of the fundamental nature of operations between sets in geometry. We have been able to prove some surprising theorems that shed light on the question in the title.