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On Poincaré constants related to isoperimetric problems in convex bodies

Dorin Bucur, Ilaria Fragalà

TL;DR

The paper addresses how the geometry of a convex body governs the Poincaré constant in W^{1,1}, providing a sharp lower bound for the inverse Poincaré constant that includes a flatness-dependent term. It fuses dimension-reduction techniques with shortest-fence reformulations in BV spaces to derive a general bound σ_1(Ω, φ) ≥ 2/D_Ω + C_0 a_2^2 / D_Ω^3 and, in 2D, an explicit Bonnesen-type inequality involving the inradius ρ_Ω, namely σ_1(Ω) ≥ 2/D_Ω + (4/3) ρ_Ω^2 / D_Ω^3, with sharp asymptotics realized by ball-strip intersections. A key intermediate result is that balls maximize the Poincaré constant among convex bodies of fixed width, guiding the 2D analysis where constant-width optimization reduces to a ball. The work advances the understanding of how domain flatness shapes spectral-type constants and furnishes tools for sharp geometric-variational inequalities in convex geometry.

Abstract

For any convex set $Ω\subset {\mathbb R} ^N$, we provide a lower bound for the inverse of the Poincaré constant in $W ^ {1, 1}(Ω)$: it refines an inequality in terms of the diameter due to Acosta-Duran, via the addition of an extra term giving account for the flatness of the domain. In dimension $N = 2$, we are able to make the extra term completely explicit, thus providing a new Bonnesen-type inequality for the Poincaré constant in terms of diameter and inradius. Such estimate is sharp, and it is asymptotically attained when the domain is the intersection of a ball with a strip bounded by parallel straight lines, symmetric about the centre of the ball. As a key intermediate step, we prove that the ball maximizes the Poincaré constant in $W ^ {1, 1} (Ω)$, among convex bodies $Ω$ of given constant width.

On Poincaré constants related to isoperimetric problems in convex bodies

TL;DR

The paper addresses how the geometry of a convex body governs the Poincaré constant in W^{1,1}, providing a sharp lower bound for the inverse Poincaré constant that includes a flatness-dependent term. It fuses dimension-reduction techniques with shortest-fence reformulations in BV spaces to derive a general bound σ_1(Ω, φ) ≥ 2/D_Ω + C_0 a_2^2 / D_Ω^3 and, in 2D, an explicit Bonnesen-type inequality involving the inradius ρ_Ω, namely σ_1(Ω) ≥ 2/D_Ω + (4/3) ρ_Ω^2 / D_Ω^3, with sharp asymptotics realized by ball-strip intersections. A key intermediate result is that balls maximize the Poincaré constant among convex bodies of fixed width, guiding the 2D analysis where constant-width optimization reduces to a ball. The work advances the understanding of how domain flatness shapes spectral-type constants and furnishes tools for sharp geometric-variational inequalities in convex geometry.

Abstract

For any convex set , we provide a lower bound for the inverse of the Poincaré constant in : it refines an inequality in terms of the diameter due to Acosta-Duran, via the addition of an extra term giving account for the flatness of the domain. In dimension , we are able to make the extra term completely explicit, thus providing a new Bonnesen-type inequality for the Poincaré constant in terms of diameter and inradius. Such estimate is sharp, and it is asymptotically attained when the domain is the intersection of a ball with a strip bounded by parallel straight lines, symmetric about the centre of the ball. As a key intermediate step, we prove that the ball maximizes the Poincaré constant in , among convex bodies of given constant width.

Paper Structure

This paper contains 10 sections, 17 theorems, 258 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The geometric side of optimal Poincaré constants in $W ^{ 1, 1}$
  3. Proof of Theorem \ref{['t:1gap']}
  4. Lower bound for the area proportion of cells
  5. One-dimensional refined Poincaré inequality
  6. Proof of Theorem \ref{['t:constwidth']}
  7. Proof of Theorem \ref{['t:newexplicit']}
  8. Appendix
  9. Asymptotics of the constant $\mathcal{K}_0$ as $p \to 1^+$
  10. Proofs of the results of Section \ref{['sec:fence']}

Key Result

Theorem 1

Let $N\ge 2$ and $m \in \mathbb{N} \setminus \{ 0 \}$. There exists a positive constant $C_0 = C_0(N , m)$ such that, for every open bounded convex domain $\Omega$ in $\mathbb{R} ^N$ with diameter $D_\Omega$ and John ellipsoid of semi-axes $a_1 \geq \dots \geq a_N$, and every $(\frac{1}{m})$-concave

Figures (2)

  • Figure 1: The geometry of Proposition \ref{['p:cruciall']}: the dotted curves represent portions of $\partial \Omega$; the coulored regions above and below $\vec{x}$ represent respectively $\Delta _\varepsilon ^+$ and $\Delta _\varepsilon ^ -$.
  • Figure 2: The geometry of Case 2

Theorems & Definitions (22)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Remark 5
  • Proposition 6
  • Proposition 7
  • Corollary 8
  • Remark 9
  • Proposition 10
  • ...and 12 more