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A geometric approach to Poincaré inequality and Minkowski content of separating sets

Emanuele Caputo, Nicola Cavallucci

Abstract

The goal of this paper is to continue the study of the relation between the Poincaré inequality and the lower bounds of Minkowski content of separating sets, initiated in our previous work [Caputo, Cavallucci: Poincaré inequality and energy of separating sets, arXiv 2401.02762]. A new shorter proof is provided. An intermediate tool is the study of the lower bound of another geometric quantity, called separating ratio. The main novelty is the description of the relation between the infima of the separating ratio and the Minkowski content of separating sets. We prove a quantitative comparison between the two infima in the local quasigeodesic case and equality in the local geodesic one. No Poincaré assumption is needed to prove it. The main tool employed in the proof is a new function, called the position function, which allows in a certain sense to fibrate a set in boundaries of separating sets. We also extend the proof to measure graphs, where due to the combinatorial nature of the problem, the approach is more intuitive. In the appendix, we revise some classical characterizations of the p-Poincaré inequality, by proving along the way equivalence with a notion of p-pencil that extends naturally the definition for p = 1.

A geometric approach to Poincaré inequality and Minkowski content of separating sets

Abstract

The goal of this paper is to continue the study of the relation between the Poincaré inequality and the lower bounds of Minkowski content of separating sets, initiated in our previous work [Caputo, Cavallucci: Poincaré inequality and energy of separating sets, arXiv 2401.02762]. A new shorter proof is provided. An intermediate tool is the study of the lower bound of another geometric quantity, called separating ratio. The main novelty is the description of the relation between the infima of the separating ratio and the Minkowski content of separating sets. We prove a quantitative comparison between the two infima in the local quasigeodesic case and equality in the local geodesic one. No Poincaré assumption is needed to prove it. The main tool employed in the proof is a new function, called the position function, which allows in a certain sense to fibrate a set in boundaries of separating sets. We also extend the proof to measure graphs, where due to the combinatorial nature of the problem, the approach is more intuitive. In the appendix, we revise some classical characterizations of the p-Poincaré inequality, by proving along the way equivalence with a notion of p-pencil that extends naturally the definition for p = 1.
Paper Structure (19 sections, 25 theorems, 116 equations, 4 figures)

This paper contains 19 sections, 25 theorems, 116 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The position function and the proof of Theorem \ref{['thm:main_equivalence_minima']}
  3. Structure of the paper
  4. Acknowledgments
  5. Preliminaries
  6. Curves in metric spaces
  7. Connectivity properties
  8. Measure properties and Poincaré inequality
  9. Minkowski content
  10. The $p$-set-connectedness and the $p$-set-pencil
  11. The proof of Theorem \ref{['theo:main-intro-p=1']}
  12. Relation between Minkowski content and separating ratio
  13. Path separating sets
  14. The position function
  15. Equivalence of minima via the position function
  16. ...and 4 more sections

Key Result

Theorem 1.4

Let $({\rm X},{\sf d},\mathfrak m)$ be a doubling metric measure space which is path connected and locally $\Lambda$-quasiconvex. Let $x,y \in {\rm X}$. Then the following conditions are quantitatively equivalent:

Figures (4)

  • Figure 1: The picture gives an informal explanation of the proof of the Theorem in the the toy example of the two dimensional Euclidean case for a specific choice of $x,y$ and $D$ in the definition of separating ratio.
  • Figure 2: Let us consider $\gamma$ and $A \subseteq \mathbb{R}^2$ as in the picture. In such a case the width of $A$ with respect to $x,y$ is equal to $\delta_1+\delta_2$ and in particular is realized by the straight curve connecting $x$ to $y$. The function ${\rm pos}_A((\gamma \cap A)_\tau)$ computes the position of the point reached when the curve travels a piece of length $\tau$ inside $A$. Its graph is reported in the picture. In this case the level set of the position function ${\rm pos}_A$ fibrates the set $A$ by straight vertical lines.
  • Figure 3: For the given set $A$ made of red points, the point $z_1$ has position $1$, i.e. ${{\rm pos}_A}(z_1)=1$, while ${\rm pos}_A(z_2)=2$. Indeed $z_1$ has position $2$ with respect to the blue path, but it has position $1$ with respect to the green one. Instead there are no paths for which $z_2$ is the first intersection point with $A$.
  • Figure 4: The construction of the sets $A_i$'s as in the proof.

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 2.1: BeerGarrido2015
  • Lemma 2.2
  • proof
  • Lemma 2.3: Hei01, CaputoCavallucci2024
  • ...and 47 more