Three quantitative versions of the P\'al inequality

TL;DR

This paper develops three quantitative versions of the Pál inequality, linking how close a convex set's area is to the minimum with its geometric similarity to an equilateral triangle, and also introduces a new inequality for inradius.

Contribution

It introduces three new quantitative inequalities related to the Pál inequality and a novel inradius inequality under minimal width constraints.

Findings

Quantitative bounds relating area closeness to geometric similarity.

Three new inequalities providing stability versions of the Pál inequality.

A new inequality for the inradius of convex sets with minimal width.

Abstract

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.