Convex Equipartitions inspired by the little cubes operad

Abstract

A decade ago two groups of authors, Karasev, Hubard and Aronov, and Blagojevi\'c 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 $n_1$ parts, then each part into $n_2$ parts, and so on. In this way we obtain iterated convex equipartions of a given convex body into $n=n_1...n_k$ 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 \& 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 $n_i$'s are powers of the same prime. In particular, for the failure of the scheme outside prime power case we give three different proofs.