# Curve shortening makes convex curves circular

```@article{Gage1984CurveSM,
title={Curve shortening makes convex curves circular},
author={Michael E. Gage},
journal={Inventiones mathematicae},
year={1984},
volume={76},
pages={357-364}
}```
• M. Gage
• Published 1 June 1984
• Medicine
• Inventiones mathematicae
269 Citations
• Physics
• 2020
We present a numerical method for computing the evolution of a planar, star-shaped curve under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flows. This
Let \$X_0, \widetilde{X}\$ be two smooth, closed and locally convex curves in the plane with same winding number. A curvature flow with a nonlocal term is constructed to evolve \$X_0\$ into
• Mathematics
• 2016
Mimicking Andrews-Bryan’s argument, it is proved in this note that Gage’s original normalized curve shortening flow can also yield the Grayson theorem.
• Mathematics
• 2014
In this note we show a variational proof of Matthew Grayson’s convexification theorem for simple closed curves moving by curvature in the plane. CONTENTS
We provide a direct proof of a noncollapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial
We provide a direct proof of a non-collapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial
• Mathematics
• 2009
It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they
We show that an analog of the Gage-Grayson-Hamilton Theorem for curves moving according to their mean curvature holds for the motion of quadrilaterals according to their Menger curva- ture.
A set of conditions on the local validity of a medial axis transform and a differential equation for the change of smooth parts of the medial axis when its generating curve evolves under MCM are presented.

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Because of Property 1, any Bonnesen inequality implies the isoperimetric inequality (1). From Property 2, it follows that equality can hold in (1) only when C is a circle. The effect of Property 3 is
Fundamentals. Hyperplanes. Helly-Type Theorems. Kirchberger-Type Theorems. Special Topics in E2. Families of Convex Sets. Characterizations of Convex Sets. Polytopes. Duality. Optimization. Convex