In analytic geometry, the isoperimetric ratio of a simple closed curve in the Euclidean plane is the ratio L2/A, where L is the length of the curve and A is its area.
It is a dimensionless quantity that is invariant under similarity transformations of the curve.
The curve-shortening flow decreases the isoperimetric ratio of any smooth convex curve so that, in the limit as the curve shrinks to a point, the ratio becomes 4π.
[2] For higher-dimensional bodies of dimension d, the isoperimetric ratio can similarly be defined as Bd/Vd − 1 where B is the surface area of the body (the measure of its boundary) and V is its volume (the measure of its interior).
[3] Other related quantities include the Cheeger constant of a Riemannian manifold and the (differently defined) Cheeger constant of a graph.