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
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References

SHOWING 1-3 OF 3 REFERENCES

BONNESEN-STYLE ISOPERIMETRIC INEQUALITIES

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

Convex Sets and Their Applications

Fundamentals. Hyperplanes. Helly-Type Theorems. Kirchberger-Type Theorems. Special Topics in E2. Families of Convex Sets. Characterizations of Convex Sets. Polytopes. Duality. Optimization. Convex