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Eight-Partitioning Points in 3D, and Efficiently Too

Boris Aronov, Abdul Basit, Indu Ramesh, Gianluca Tasinato, Uli Wagner

TL;DR

The paper proves a topological Hadwiger-type result in $\mathbb{R}^3$ guaranteeing an eight-partition whose two planes intersect along a line in a prescribed direction, and extends the result to finite point sets via limiting arguments. It develops a two-step, $G$-equivariant framework, leveraging a zero of a map on $\mathbb{S}^1\times\mathbb{S}^3$ to certify existence, aided by a 2D four-partition lemma. It then translates the topological insight into an efficient algorithm that computes such eight-partitions for $n$ points in time $O^{*}(n^{7/3})$ by reducing to a median-level intersection problem in plane arrangements and exploiting dynamic ray-shooting. The work connects Hadwiger's and Grünbaum's problems with computational geometry through duality and level complexity, yielding both existential and constructive outcomes with implications for 3D mass and point partitions.

Abstract

An {\em eight-partition} of a finite set of points (respectively, of a continuous mass distribution) in $\mathbb{R}^3$ consists of three planes that divide the space into $8$ octants, such that each open octant contains at most $1/8$ of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in $\mathbb{R}^3$ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: Any mass distribution (or point set) in $\mathbb{R}^3$ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of $n$ points in~$\mathbb{R}^3$ (with prescribed normal direction of one of the planes) in time $O^{*}(n^{7/3})$.

Eight-Partitioning Points in 3D, and Efficiently Too

TL;DR

The paper proves a topological Hadwiger-type result in guaranteeing an eight-partition whose two planes intersect along a line in a prescribed direction, and extends the result to finite point sets via limiting arguments. It develops a two-step, -equivariant framework, leveraging a zero of a map on to certify existence, aided by a 2D four-partition lemma. It then translates the topological insight into an efficient algorithm that computes such eight-partitions for points in time by reducing to a median-level intersection problem in plane arrangements and exploiting dynamic ray-shooting. The work connects Hadwiger's and Grünbaum's problems with computational geometry through duality and level complexity, yielding both existential and constructive outcomes with implications for 3D mass and point partitions.

Abstract

An {\em eight-partition} of a finite set of points (respectively, of a continuous mass distribution) in consists of three planes that divide the space into octants, such that each open octant contains at most of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: Any mass distribution (or point set) in admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of points in~ (with prescribed normal direction of one of the planes) in time .
Paper Structure (8 sections, 20 theorems, 7 equations, 1 figure)

This paper contains 8 sections, 20 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The topological result
  3. Notation and preliminaries
  4. The main topological result
  5. Levels in arrangements of planes
  6. The algorithm
  7. Limit arguments
  8. Four-partitioning in the plane with a prescribed bisector

Key Result

Theorem 1

Let $\mu$ be a mass distribution on $\mathbb{R}^3$, and let $v \in \mathbb{S}^2$. Then there exists a triple of planes $(H_1, H_2, H_3)$ that form an eight-partition for $\mu$ and such that the normal vector of $H_1$ is $v$.

Figures (1)

  • Figure 2: Up to symmetries, the different possibilities for the image under $\pi$ of a grid square $S$, which is always always a grid curve in $\mathbb{Z}^2$, by Lemma \ref{['fact:grid-to-grid']}. Note that the image cannot have odd winding number.

Theorems & Definitions (23)

  • Theorem 1: HYDEP
  • Theorem 2: blag_karasev_2016
  • Theorem 3
  • Theorem 4: Algorithm
  • Theorem 6
  • Lemma 6: Four-partitioning a mass distribution in $\bbR^2$ blag_karasev_2016
  • Theorem 7
  • Theorem 11: Computation of an eight-partition
  • Remark
  • Lemma 12
  • ...and 13 more