Isoperimetric ratio

In analytic geometry, the isoperimetric ratio of a simple closed curve in the Euclidean plane is the ratio , where is the length of the curve and is its area. It is a dimensionless quantity that is invariant under similarity transformations of the curve.
According to the isoperimetric inequality, the isoperimetric ratio has its minimum value, 4, for a circle; any other curve has a larger value. Thus, the isoperimetric ratio can be used to measure how far from circular a shape is.
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 4.
For higher-dimensional bodies of dimension "d", the isoperimetric ratio can similarly be defined as where "B" is the surface area of the body (the measure of its boundary) and "V" is its volume (the measure of its interior). Other related quantities include the Cheeger constant of a Riemannian manifold and the (differently defined) Cheeger constant of a graph.