Right. And if you’ve got three centroids, you’ve got three points that uniquely define a plane. Since this plane ipso facto passes through the centroid of each shape, the plane bisects each volume! No calculus, topology, or higher geometry required. So why does Intermediate Value Theorem / Borsuk-Ulam come into it at all?
I suppose that at some point in proving the existence of centroids, you would probably use I.V.T., but it seems sloppy to involve I.V.T. in the traditional proof of Ham Sandwich Theorem when you could just offload it onto the existence of centroids as a lemma. As for higher dimensional cases, is it possible that centroids are not generally well-defined?
