  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 abstract 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 investigate two types of subsets. The first type are linear subspaces of
$\mathbb{R}^d$, i.e., $k$-dimensional flats containing the origin. We show that
for any continuous assignment of $d$ mass distributions to the $k$-dimensional
linear subspaces of $\mathbb{R}^d$, there is always a subspace on which we can
simultaneously bisect the images of all $d$ assignments. We extend this result
to center transversals, a generalization of Ham-Sandwich cuts. As for
Ham-Sandwich cuts, we further show that for $d-k+2$ masses, we can choose $k-1$
of the vectors defining the $k$-dimensional subspace in which the solution
lies.
  The second type of subsets we consider are subsets that are determined by
families of $n$ hyperplanes in $\mathbb{R}^d$. Also in this case, we find a
Ham-Sandwich-type result. In an attempt to solve a conjecture by Langerman
about bisections with several cuts, we show that our underlying topological
result can be used to prove this conjecture in a relaxed setting.
