A counterexample to the Macbeath point inclusion

The center of a largest centrally symmetric subset of a convex body need not satisfy the usual dimension bound on asymmetry. We compute this center for a square pyramid in $\mathbb R^3$ and show that reflection about it, followed by shrinking by a factor of three, takes the apex outside the pyramid. This gives a counterexample to Ivanov's Macbeath point inclusion conjecture.