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<p>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 <i>quantitative</i> 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.</p>

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Three Quantitative Versions of the Pál Inequality

  • 期刊论文

  • 发表日期：

    2025年2月15日

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  • Ilaria Lucardesi,
  • Davide Zucco

摘要

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.