The isoperimetric quotient of a convex body decreases monotonically under the Eikonal abrasion model

TL;DR

This paper proves that when a convex shape undergoes Eikonal abrasion with uniform inward normal speed, its isoperimetric quotient consistently decreases over time, revealing a monotonic geometric property.

Contribution

It establishes the monotonic decrease of the isoperimetric quotient for convex bodies under the Eikonal abrasion model, a novel geometric insight.

Findings

Isoperimetric quotient decreases monotonically during Eikonal abrasion

Convex shapes become less efficient in enclosing volume relative to surface area over time

Provides a new understanding of shape evolution under uniform normal speed abrasion

Abstract

We show that under the Eikonal abrasion model, prescribing uniform normal speed in the direction of the inward surface normal, the isoperimetric quotient of a convex shape is decreasing monotonically.