Convex equipartitions inspired by the little cubes operad
摘要
A decade ago two groups of authors, Karasev, Hubard and Aronov and Blagojević and Ziegler, have shown that the regular convex partitions of a Euclidean space into n parts yield a solution to the generalised Nandakumar and Ramana-Rao conjecture when n is a prime power. This was obtained by parametrising the space of regular equipartitions of a given convex body with the classical configuration space.
Now, we repeat the process of regular convex equipartitions many times, first partitioning the Euclidean space into n1 parts, then each part into n2 parts, and so on. In this way we obtain iterated convex equipartions of a given convex body into n = n1 ⋯ nk parts. Such iterated partitions are parametrised by the (wreath) product of classical configuration spaces. We develop a new configuration space–test map scheme for solving the generalised Nandakumar and Ramana-Rao conjecture using the Hausdorff metric on the space of iterated convex equipartions.
The new scheme yields a solution to the conjecture if and only if all the ni’s are powers of the same prime. In particular, for the failure of the scheme outside prime power case we give three different proofs.