An isoperimetric inequality with applications to curve shortening (Q790436): Difference between revisions

For closed convex \(C^ 2\) curves in the plane with length L, area A and curvature function \(\kappa\), the inequality \(\pi \frac{L}{A}\leq \int^{L}_{O}\kappa^ 2ds\) is proved. It is used to show the following: When a convex curve is deformed along its (inner) normal at a rate proportional to its curvature, then the isoperimetric ratio \(L^ 2/A\) decreases.