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Variations on the capacitary inradius

Francesco Bozzola, Lorenzo Brasco

TL;DR

The paper develops a capacitary generalization of the inradius for open subsets of $\mathbb{R}^N$, denoted $R_{p,\gamma}(\Omega)$, integrating $p$-capacity to capture fatness while discounting capacity-zero removals and linking to Poincaré-type inequalities. It systematically compares variants based on different shape metrics and different notions of capacity (Maz'ya-Shubin and Gallagher), establishing two-sided estimates and highlighting when capacitary radii coincide with the classical inradius $r_\Omega$ under a simple measure-density condition. The main result provides explicit, quantitative bounds $r_\Omega \le R_{p,\gamma}(\Omega) \le \mathcal{C} r_\Omega$ under this density hypothesis, with consequences for sharp Poincaré–Sobolev constants and Cheeger-type inequalities, including a Buser-type bound for the Dirichlet Laplacian. The work also connects several capacitary notions (absolute/relative and endpoint cases) and demonstrates the applicability of the theory to domains with power-like cusps, via a tangible geometric condition such as a uniform exterior funnel. Overall, the paper offers a robust framework to translate geometric fatness into spectral and functional-analytic estimates for broad classes of domains.

Abstract

We discuss some properties of the capacitary inradius for an open set. This is an extension of the classical concept of inradius (i.e. the radius of a largest inscribed ball), which takes into account capacitary effects. Its introduction dates back to the pioneering works of Vladimir Maz'ya. We present some variants of this object and their mutual relations, as well as their connections with Poincaré inequalities. We also show that, under a mild regularity assumption on the boundary of the sets, the capacitary inradius is equivalent to the classical inradius. This comes with an explicit estimate and it permits to get a Buser-type inequality for a large class of open sets, whose boundaries may have power-like cusps of arbitrary order. Finally, we present a couple of open problems.

Variations on the capacitary inradius

TL;DR

The paper develops a capacitary generalization of the inradius for open subsets of , denoted , integrating -capacity to capture fatness while discounting capacity-zero removals and linking to Poincaré-type inequalities. It systematically compares variants based on different shape metrics and different notions of capacity (Maz'ya-Shubin and Gallagher), establishing two-sided estimates and highlighting when capacitary radii coincide with the classical inradius under a simple measure-density condition. The main result provides explicit, quantitative bounds under this density hypothesis, with consequences for sharp Poincaré–Sobolev constants and Cheeger-type inequalities, including a Buser-type bound for the Dirichlet Laplacian. The work also connects several capacitary notions (absolute/relative and endpoint cases) and demonstrates the applicability of the theory to domains with power-like cusps, via a tangible geometric condition such as a uniform exterior funnel. Overall, the paper offers a robust framework to translate geometric fatness into spectral and functional-analytic estimates for broad classes of domains.

Abstract

We discuss some properties of the capacitary inradius for an open set. This is an extension of the classical concept of inradius (i.e. the radius of a largest inscribed ball), which takes into account capacitary effects. Its introduction dates back to the pioneering works of Vladimir Maz'ya. We present some variants of this object and their mutual relations, as well as their connections with Poincaré inequalities. We also show that, under a mild regularity assumption on the boundary of the sets, the capacitary inradius is equivalent to the classical inradius. This comes with an explicit estimate and it permits to get a Buser-type inequality for a large class of open sets, whose boundaries may have power-like cusps of arbitrary order. Finally, we present a couple of open problems.

Paper Structure

This paper contains 13 sections, 20 theorems, 209 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Description of the results
  4. Plan of the paper
  5. Preliminaries
  6. Different metrics
  7. Different capacities
  8. Maz'ya-Shubin inradius
  9. Gallagher inradius
  10. Capacitary inradius and measure density condition
  11. A two-sided estimate
  12. Some consequences
  13. Example: uniform exterior funnel condition

Key Result

Lemma 2.2

Let $E\subseteq\mathbb{R}^N$ be an open set and let $\Sigma\subseteq E$ be a compact set. For every $1\le p<\infty$ we have

Figures (2)

  • Figure 1: A cube $Q_\ell(\mathbf{p}_i)$, containing a concentric ball with radius $2\,r_\Omega$. The black dot indicates a point outside $\Omega$. By construction, at this point we can center a ball which is feasible for the measure density index $\theta_{\Omega,r_0}(t)$.
  • Figure 2: In bold line, the profile of a $\beta-$funnel, with $\beta=2/5$, opening $\delta=1/2$ and height $h_0=1/2$. The circle in dashed line has radius $r_0$ given by \ref{['raggiobeta']}.

Theorems & Definitions (57)

  • Remark 1.1
  • Remark 1.2: The case $p>N$
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 47 more