The Ham-Sandwich theorem is a well-known result in geometry. It states that any $d$ mass distributions in $\mathbb{R}^d$ can be simultaneously bisected by a hyperplane. The result is tight, that is, there are examples of $d+1$ mass distributions that cannot be simultaneously bisected by a single hyperplane. In this talk we will study the following question: given a continuous assignment of mass distributions to certain subsets of $\mathbb{R}^d$, is there a subset on which we can bisect more masses than what is guaranteed by the Ham-Sandwich theorem?

We will study two different types of subsets, motivated by conjectures by Luis Barba (which we will answer) and Stefan Langerman (which we solve only in a relaxed setting), respectively. Some of the results we also extend to center transversals, a generalization of Ham-Sandwich cuts.

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