  In this paper, we construct an injection $A \times B \rightarrow M \times M$
from the product of any two nonempty subsets of the symmetric group into the
square of their midpoint set, where the metric is that corresponding to the
conjugacy class of transpositions. If $A$ and $B$ are disjoint, our
construction allows to inject two copies of $A \times B$ into $M \times M$.
These injections imply a positively curved Brunn-Minkowski inequality for the
symmetric group analogous to that obtained by Ollivier and Villani for the
hypercube. However, while Ollivier and Villani's inequality is optimal, we
believe that the curvature term in our inequality can be improved. We identify
a hypothetical concentration inequality in the symmetric group and prove that
it yields an optimally curved Brunn-Minkowski inequality.
