A lower bound for the isoperimetric deficit

TL;DR

This paper establishes a Bonnesen-style inequality providing a lower bound for the isoperimetric deficit of convex curves, along with a geometric interpretation of the equality case, enhancing understanding of curve geometry.

Contribution

It introduces a new inequality linking the isoperimetric deficit to geometric invariants of convex curves and offers a geometric interpretation for the equality condition.

Findings

Derived a lower bound for the isoperimetric deficit

Provided a geometric interpretation for the equality case

Enhanced understanding of convex curve geometry

Abstract

In this paper we provide a Bonnesen-style inequality which gives a lower bound for the isoperimetric deficit corresponding to a closed convex curve in terms of some geometrical invariants of this curve. Moreover we give a geometrical interpretation for the case when equality holds.