While the two pieces of bread and the piece of ham might have centroids, the problem here is to slice them simultaneously so that all three are bisected with a single slice, without moving them.
When I teach the Ham Sandwich Theorem I usually emphasize that the three volumes can be anywhere, for example one piece of bread on your kitchen counter, the second piece of bread in orbit around the moon, and the ham a chunk the size, shape, and location of the star Betelgeuse. There exists a single plane that slices all of them simultaneously.
By the way, I’m pretty sure I saw a video (we called it a “movie” back then) on the Ham Sandwich and the Hairy Billiard Ball Theorems when I was an undergraduate in the mid-1970s. I don’t know where that video is today. Funny theorem names seem to attract popularizations now as well as they did then.
