Three quantitative versions of the Pál inequality

TL;DR

This paper provides three quantitative refinements of Pál's inequality for planar convex bodies of fixed minimal width, each bounding the area deficit $|K|/\omega^2(K) - 1/\sqrt{3}$ from below by a distance-to-equilateral-triangle measure. It proves a Bonnesen-type bound via the inradius $r(K)$, a Hausdorff-distance bound to an optimal equilateral triangle, and a Fraenkel-asymmetry bound, all with explicit constants $c_1=1/\sqrt{5}$, $c_2=1/(25\sqrt{5})$, and $c_3=1/(25(3\sqrt{3}+2)\sqrt{5})$, respectively, and establishes the sharpness of the exponent 1. A novel quantitative inequality for the inradius under a minimal-width constraint is also derived, using a truncated asymmetry $\beta_E(K)=\min\{\alpha(K),1/6\}$ and yielding a concrete constant. The methods combine circumscribed-triangle geometry, inradius–width relations, and careful analysis of Hausdorff and Fraenkel asymmetries, culminating in an equivalence between Hausdorff and Fraenkel measures in this setting. Collectively, these results deepen the stability portrait of the Pál inequality and have potential implications for geometric tomography and related areas of convex geometry.

Abstract

The Pál inequality is a classical result which asserts that among all planar convex sets of given width the equilateral triangle is the one of minimal area. In this paper we prove three quantitative versions of this inequality, by quantifying how the closeness of the area of a convex set, of certain width, to the minimal value implies its closeness to the equilateral triangle. As a by-product, we also present a novel result concerning a quantitative inequality for the inradius of a set, under minimal width constraint.

Figures (6)

• Figure 1: In bold it is shown the boundary $\partial K$ of $K$. The circle is the indisk $D$ of $K$ and the triangle $V_1V_2V_3$ is the circumscribed triangle $T_K$. The dashed lines are parallel to $\ell_i$ at distance $\omega_i$.
• Figure 2: The triangles $T$ and $E$ of Lemma \ref{['lem.triangle']}.
• Figure 3: The two similar triangles $V_1V_2V_3$ and $V_5V_4V_3$.
• Figure 4: The function $\varphi$ and its derivative $\varphi'$.
• Figure 5: The functions $f$ and $g$.

Theorems & Definitions (41)

• Theorem 1.1: Quantitative Pál inequality via inradius
• Theorem 1.2: Quantitative Pál inequality via Hausdorff distance
• Theorem 1.3: Quantitative Pál inequality via Fraenkel asymmetry
• Remark 1.4: Sharpness of the exponents and computability of the constants
• Lemma 2.1: Contact points
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4