During my Christmas holidays I went through the regular ritual of eating cake with family relations. Occasionally I would observe crumbs fall off the sides of the cake and then think of the relative share each person got. These rituals continued until for the first time, I found an interesting math problem based on the challenge of dividing cake into equal volumes without leaving any crumbs. In fact it led me to an interesting conjecture at the intersection of geometry, topology and measure theory.
A normal cake can be considered a compact euclidean manifold homeomorphic to the sphere. If this manifold is also convex:
For any point on the boundary of our manifold, there exists a hyperplane through this point such that we may decompose our manifold into three path-connected components that satisfy:
The challenge is to show that the above volume-splitting properties hold for any compact manifold that is also convex but doesn’t hold otherwise. By this I mean that if our manifold happens to be non-convex then the above properties don’t hold for every point on its boundary. In other words we would observe crumbs.
This might seem like a relatively simple problem but it’s actually a more complex variant of the Ham Sandwich theorem which I learned from other users of the math stackexchange when I first shared the problem.
I leave it as an exercise to the reader to show that the volume-splitting property holds for any compact and convex manifold. The hard part is showing that for any compact and non-convex manifold the volume-splitting property doesn’t hold for all points on the boundary.
Right now it’s not clear to me whether this problem might exist under another name. If it happens to be unsolved I promise ten cupcakes to the first person that manages to solve it before me.
Note: I have written a more structured article based on this blog post here.