  We introduce a concept that generalizes several different notions of a
"centerpoint" in the literature. We develop an oracle-based algorithm for
convex mixed-integer optimization based on centerpoints. Further, we show that
algorithms based on centerpoints are "best possible" in a certain sense.
Motivated by this, we establish several structural results about this concept
and provide efficient algorithms for computing these points. Our main
motivation is to understand the complexity of oracle based convex mixed-integer
optimization.
