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Cheeger's isoperimetric problem for Gaussian mixtures

Lukas Liehr

Abstract

In any dimension $n$, we determine the Cheeger constant and the Cheeger sets of the Gaussian mixture $μ(x) = pγ(x-a) + (1-p)γ(x-b)$, where $p \in [0,1]$, $a,b \in \mathbb{R}^n$, and $γ: \mathbb{R}^n \to (0,\infty)$ denotes a Gaussian. In particular, we characterize the Cheeger sets for $μ$ in terms of specific half-spaces perpendicular to $a-b$, thereby confirming the conjectured solution to the Cheeger problem for Gaussian mixtures. Finally, we study the regime of parameters $p,a,b$ in which $μ$ admits a unique Cheeger set.

Cheeger's isoperimetric problem for Gaussian mixtures

Abstract

In any dimension , we determine the Cheeger constant and the Cheeger sets of the Gaussian mixture , where , , and denotes a Gaussian. In particular, we characterize the Cheeger sets for in terms of specific half-spaces perpendicular to , thereby confirming the conjectured solution to the Cheeger problem for Gaussian mixtures. Finally, we study the regime of parameters in which admits a unique Cheeger set.
Paper Structure (9 sections, 14 theorems, 110 equations)

This paper contains 9 sections, 14 theorems, 110 equations.

Table of Contents

  1. Introduction and results
  2. Main results
  3. Outline and proof approach
  4. Isoperimetric inequalities
  5. Preliminaries
  6. Isoperimetric inequalities for $\mu$
  7. Local log-concavity
  8. Proof of Theorem \ref{['thm:main1']} and Theorem \ref{['thm:main2']}
  9. Proof of Theorem \ref{['thm:main3']}

Key Result

Theorem 1.1

Let $a,b\in\mathbb R^n$ with $a \neq b$, let $p\in(0,1)$, and let $\mu$ be the measure Define $d :=|a-b|$, $m := \min \{ p,1-p \}$, and $r^* > 0$ via the equality $Q_{m,d}(r^*)=\tfrac{1}{2}$. Then the Cheeger constant of $\mu$ is given by the minimum If $\mathcal{O} \subset [0,r^*]$ is the set of minimizers and if $\nu := \frac{b-a}{|b-a|}$, then the following holds:

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • ...and 17 more