A doubly monotone flow for constant width bodies in $\mathbb{R}^3$

TL;DR

This paper introduces a flow for three-dimensional constant width bodies that increases volume and decreases circumradius over time, leading to convergence to a sphere, with potential for further study in reverse time.

Contribution

It presents a novel flow in 3D constant width bodies that guarantees convergence to a sphere while controlling volume and circumradius.

Findings

Flow exists for all positive times.

Flow converges to a sphere as time approaches infinity.

Flow potentially useful for shape optimization in reverse.

Abstract

We introduce a flow in the space of constant width bodies in three-dimensional Euclidean space that simultaneously increases the volume and decreases the circumradius of the shape as time increases. Starting from any initial constant width figure, we show that the flow exists for all positive times and converges to a closed ball as time tends to plus infinity. We also anticipate that this flow is interesting to study for negative times and that it would provide a mechanism to decrease the volume and increase the circumradius of a constant width body.