Convex Equipartitions inspired by the little cubes operad

TL;DR

This paper extends the convex equipartition approach to iterated partitions parametrized by configuration spaces, providing new solutions to the Nandakumar-Ramana-Rao conjecture for prime power cases and multiple proofs of failure otherwise.

Contribution

It introduces a new configuration space and test map scheme for iterated convex equipartitions, advancing the understanding of the conjecture's validity.

Findings

Solution for prime power cases using the new scheme.

Three different proofs of failure outside prime power cases.

Extension of equipartition methods to iterated partitions.

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.