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Minimizers of the Maximum Distance Problem via an Analyst's Traveling Salesperson Algorithm

Enrique Alvarado, Silvia Ghinassi, Lisa Naples

Abstract

We provide an upper and lower bound for the length of Maximum Distance Problem minimizers in terms of a finite scale geometric square sum.

Minimizers of the Maximum Distance Problem via an Analyst's Traveling Salesperson Algorithm

Abstract

We provide an upper and lower bound for the length of Maximum Distance Problem minimizers in terms of a finite scale geometric square sum.
Paper Structure (15 sections, 17 theorems, 57 equations, 3 figures)

This paper contains 15 sections, 17 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Lower bound
  4. Upper bound
  5. Building convex hulls
  6. Initialization and first generation of construction
  7. Inductive Construction
  8. Properties of the collection of convex hulls
  9. Labeling convex hulls as good and bad
  10. Building candidate curves
  11. Initialization and first generation construction
  12. Inductive Construction
  13. Termination of construction
  14. Verifying that is a candidate curve.
  15. Estimating Length of the Candidate

Key Result

Theorem 1.4

A bounded set $E \subset \mathbb{R}^2$ is contained in a curve of finite length if and only if More precisely, if $\sum_{\substack{Q \in \mathcal{D}}} \beta^2_E(3Q) |Q|< \infty$, then there exist a curve of finite length $I$ and a universal constant $c_1>0$ so that $I \supset E$ and $\mathcal{H}^1(I) \leq c_1 \left(|E| + \sum_{\substack{Q \in \mathcal{D}}} \beta^2_E(3Q) |Q|\right)$. If $I$ is

Figures (3)

  • Figure 1: Cases (P1) and (P2) of the first generation of the construction inside $C_\varnothing$.
  • Figure 2: A comparison of $\hat{\beta}(C)$ and $\beta_E(3Q)$, for $C \in \mathscr{C}(Q)$.
  • Figure 3: Two steps of the construction of the $\Gamma_j$'s (shown in purple).

Theorems & Definitions (43)

  • Definition 1.1: Dyadic cubes
  • Definition 1.2: Hausdorff measure
  • Definition 1.3: $\beta$-numbers
  • Theorem 1.4: Analyst's Traveling Salesperson Theorem J
  • Theorem 1.5: Main theorem
  • Remark 1.6
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • ...and 33 more