Calculus of Variations and Geometric Measure Theory

I. Lucardesi - D. Zucco

Three quantitative versions of the Pál inequality

created by zucco on 29 May 2024



Inserted: 29 may 2024

Year: 2024

ArXiv: 2405.18294 PDF


The P\'al 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.