A reverse isoperimetric inequality for planar ($\alpha$, $\beta$)--convex bodies

TL;DR

This paper establishes a reverse isoperimetric inequality for planar ($ ext{α}$, $ ext{β}$)-convex bodies, identifying extremal shapes with boundaries formed by circular arcs of specified radii.

Contribution

It introduces a reverse isoperimetric inequality for ($ ext{α}$, $ ext{β}$)-convex bodies and characterizes the extremal shape with boundary arcs of two different radii.

Findings

Extremal shape has boundary composed of arcs of circles with radii α and β.

Among fixed perimeter, the extremal shape maximizes area.

The boundary configuration is explicitly characterized by circular arcs.

Abstract

In this paper, we study a reverse isoperimetric inequality for planar convex bodies whose radius of curvature is between two positive numbers 0 < $\alpha$ < $\beta$, called ($\alpha$, $\beta$)--convex bodies. We show that among planar ($\alpha$, $\beta$)--convex bodies of fixed perimeter, the extremal shape is a domain whose boundary is composed by two arcs of circles of radius $\alpha$ joined by two arcs of circles of radius $\beta$.