Published Paper

Inserted: 29 may 2024
Last Updated: 15 mar 2025

Journal: The Journal of Geometric Analysis
Volume: 35
Year: 2025
Doi: https://doi.org/10.1007/s12220-025-01931-7

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.

Keywords: convex sets, Equilateral triangle, Minimal width, Quantitative inequality