Convex Equipartitions inspired by the little cubes operad
Pavle V. M. Blagojevic, Nikola Sadovek

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 parts yield a solution to the generalised Nandakumar and Ramana-Rao conjecture when 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 parts, then each part into parts, and so on. In this way we obtain iterated convex equipartions of a given convex body into 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeochemistry and Geologic Mapping · Point processes and geometric inequalities
