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Cookie cutters: Bisections with fixed shapes

Patrick Schnider, Pablo Soberón

TL;DR

The paper studies simultaneous bisection of $d+1$ mass distributions in $\mathbb{R}^d$ using scaled copies of a fixed cookie cutter $C$, distinguishing between homothetic copies and scaled isometric copies. Using equivariant topology and Borsuk--Ulam-type theorems, it proves existence results for smooth star-shaped cutters, cylinders, and hypercubes, and extends to non-smooth and non-star-shaped cutters via new topological lemmas. Key contributions include a general framework for pairing fixed shapes with mass-partition requirements, plus concrete positive results for cylinders, hypercubes, and their non-smooth generalizations. The methods provide a foundation for further exploration of fixed-shape mass partitions and offer potential algorithmic insights despite their non-constructive nature.

Abstract

In a mass partition problem, we are interested in finding equitable partitions of smooth measures in $\mathbb{R}^d$. In this manuscript, we study the problem of finding simultaneous bisections of measures using scaled copies of a prescribed set $K$. We distinguish the problem when we are allowed to use scaled and translated copies of $K$ and the problem when we are allowed to use scaled isometric copies of $K$. These problems have only previously been studied if $K$ is a half-space or a Euclidean ball. We obtain positive results for simultaneous bisection of any $d+1$ masses for star-shaped compact sets $K$ with non-empty interior, where the conditions on the problem depend on the smoothness of the boundary of $K$. Additional proofs are included for particular instances of $K$, such as hypercubes and cylinders, answering positively a conjecture of Soberón and Takahashi. The proof methods are topological and involve new Borsuk--Ulam-type theorems.

Cookie cutters: Bisections with fixed shapes

TL;DR

The paper studies simultaneous bisection of mass distributions in using scaled copies of a fixed cookie cutter , distinguishing between homothetic copies and scaled isometric copies. Using equivariant topology and Borsuk--Ulam-type theorems, it proves existence results for smooth star-shaped cutters, cylinders, and hypercubes, and extends to non-smooth and non-star-shaped cutters via new topological lemmas. Key contributions include a general framework for pairing fixed shapes with mass-partition requirements, plus concrete positive results for cylinders, hypercubes, and their non-smooth generalizations. The methods provide a foundation for further exploration of fixed-shape mass partitions and offer potential algorithmic insights despite their non-constructive nature.

Abstract

In a mass partition problem, we are interested in finding equitable partitions of smooth measures in . In this manuscript, we study the problem of finding simultaneous bisections of measures using scaled copies of a prescribed set . We distinguish the problem when we are allowed to use scaled and translated copies of and the problem when we are allowed to use scaled isometric copies of . These problems have only previously been studied if is a half-space or a Euclidean ball. We obtain positive results for simultaneous bisection of any masses for star-shaped compact sets with non-empty interior, where the conditions on the problem depend on the smoothness of the boundary of . Additional proofs are included for particular instances of , such as hypercubes and cylinders, answering positively a conjecture of Soberón and Takahashi. The proof methods are topological and involve new Borsuk--Ulam-type theorems.

Paper Structure

This paper contains 7 sections, 13 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bisections with cookie cutters
  3. Bisections with squares and cylinders
  4. Bisections with hypercubes
  5. Bisections with non-smooth cookie cutters
  6. Generalizing to non-star-shaped cookie cutters
  7. Conclusion

Key Result

Theorem 1.1

Let $\mu_0,\ldots,\mu_{d}$ be $d+1$ mass distributions on $\mathds{R}^d$ and let $C$ be a smooth cookie cutter. Then there exists a homothetic copy $C'$ of $C$ such that $\mu_i(C')=\frac{1}{2}\mu_i(\mathds{R}^d)$ for all $i\in\{0,\ldots,d\}$.

Figures (2)

  • Figure 1: This figure describes how $C(v)$ changes as $v$ moves in $B^d$ towards the boundary. Within $(1/2)B^d$, the set $C(v)$ is a homothetic translated copy of $C$. When $v=v_2$, the magnitude of $v$ is $1/2$, and $C(v)$ is a half-space orthogonal to $n(u)$. As we keep increasing the magnitude of $v$, we change the direction of the half-space $C(v)$ until it is orthogonal to $u$ (and points in the direction of $-u$.
  • Figure 2: In illustration of the rotation of the cookie cutters.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • proof : Proof of \ref{['thm:smooth_star']}
  • Theorem 3.1
  • Corollary 3.2
  • proof : Proof of \ref{['thm:cylinders']}
  • Theorem 4.1: FrickAndFriends, Thm. 1.1
  • ...and 10 more