Cannons at sparrows (Q2352817)

This fascinating and entertaining article begins with an apparently simple problem in convex geometry, due to R. Nandakumar and R. Ramana Rao: given a convex shape, can it be partitioned into \(N\) pieces so that all pieces have equal area and perimeter? (This is different, except when \(N=2\), from the ``cake and icing problem'' of dividing the region into pieces that each have the same area and the same portion of the original body's perimeter.) The \(N=2\) case can be solved easily (if unsportingly) by the use of the Borsuk-Ulam theorem. This paper summarizes further progress on the problem, which involves even more diverse heavy artillery. To even represent the topological space of equal-area partitions requires ideas from the theory of \textit{optimal transport}, and particularly a 1938 theorem of Kantorovich on weighted Voronoi diagrams. Once posed in this way, the problem can be converted into a problem in equivariant algebraic topology via the ``configuration space / test map scheme''. If there is \textit{no} solution to the original problem, then there is an equivariant map from a certain (\(n-1\))-dimensional cell complex to the (\(n-2\))-sphere. The existence or nonexistence of this map can be determined using equivariant obstruction theory. This in turn leads, via permutahedra, to a question (solved over a hundred years ago) about prime factors of entries in Pascal's triangle, and the final result of the paper, due to the author and Blagojević: if \(N\) is a prime power, the desired dissection exists. The paper is clear and well-written, though I was puzzled to find ``equivariant obstruction theory'' abbreviated as ``EOS'' twice. This appears to be a typo: it's ``EOT'' on two other occasions.