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Ratio of Intrinsic Metric to Extrinsic Metric and Volume

Berk Ceylan

TL;DR

The paper defines the intrinsic-extrinsic distortion ratio $K$ for spaces embedded in a unit ball and seeks lower bounds on the maximal distortion $K_{\max}$ from area/volume information. A packing-number framework yields quantitative lower bounds on $K_{\max}$ under volume constraints, and the authors show these bounds can blow up as the allowed granularity $\delta$ shrinks. However, they also present counterexamples of spaces with bounded distortion yet arbitrarily large area, highlighting the necessity of extra structure. To salvage area-to-distortion control, they introduce triangular (and $R$-triangular) spaces, proving that sufficiently large area forces distortion $K\ge R/\delta$ in these settings, and discuss counting of large-distortion geodesic pairs via improved packing arguments. The results illuminate when intrinsic metrics constrain extrinsic distortion and when such bounds fail in general, guiding how to impose robust geometric hypotheses.

Abstract

We study the relationship between the ratio of intrinsic to extrinsic metrics and area. For certain surfaces inside unit ball in R3 we give lower bound on the maximum of ratio in terms of its area. We also give examples to show non-existence of global lower bounds.

Ratio of Intrinsic Metric to Extrinsic Metric and Volume

TL;DR

The paper defines the intrinsic-extrinsic distortion ratio for spaces embedded in a unit ball and seeks lower bounds on the maximal distortion from area/volume information. A packing-number framework yields quantitative lower bounds on under volume constraints, and the authors show these bounds can blow up as the allowed granularity shrinks. However, they also present counterexamples of spaces with bounded distortion yet arbitrarily large area, highlighting the necessity of extra structure. To salvage area-to-distortion control, they introduce triangular (and -triangular) spaces, proving that sufficiently large area forces distortion in these settings, and discuss counting of large-distortion geodesic pairs via improved packing arguments. The results illuminate when intrinsic metrics constrain extrinsic distortion and when such bounds fail in general, guiding how to impose robust geometric hypotheses.

Abstract

We study the relationship between the ratio of intrinsic to extrinsic metrics and area. For certain surfaces inside unit ball in R3 we give lower bound on the maximum of ratio in terms of its area. We also give examples to show non-existence of global lower bounds.

Paper Structure

This paper contains 5 sections, 8 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Spaces With Bounded Balls Yet Large Volume
  3. Families of Surfaces With Bounded $K$ But Diverging Volume
  4. Triangular Spaces
  5. Counting Geodesics With Large Distortion

Key Result

Theorem 1

Let $X\subset B^n_1$ be an intrinsic metric space for which metric is induced by $B^n_1$, where $B^n_1$ denotes unit ball in $\mathbf{R}^n$. Assume $diam(X) \geq J_n$ and $a_{X}(\frac{J_n}{3},m) \leq C(\frac{J_n}{3},m)$. Then if $\mathcal{H}_m(X)>C(\frac{J_n}{3},m)\cdot \beta_{B^n_1}(\delta)$ we hav

Figures (3)

  • Figure 1: Picture of $X_2$.
  • Figure 2: A triangular space.
  • Figure 3: Chord in the development is geodesic.

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 3
  • proof
  • ...and 3 more