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I. The Isoperimetric Inequality
For all closed curves in the plane with arc length L and area A,
4πA ≤ L^2
Moreover, equality holds if and only if the curve is a circle.
More Explanation of the Project
This page provides some terms and equations involved in the project.
II. The Definition of "Better"
For the plane curve problem, we normalize all curves to have constant arc length and then simply compare areas.
III. Fourier and Taylor Series: Noise
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How does one ”randomly” perturb a curve?
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We appeal to several theorems from analysis.
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Area problem: Fourier series.
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Parabola problem: Taylor series.
IV. Formula involved
The Area Problem:
The Parabola Problem:
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