Isoperimetric ratio

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.

According to the isoperimetric inequality, the isoperimetric ratio has its minimum value, 4π, for a circle; any other curve has a larger value.[1] 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 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.[4]

  1. ^ Berger, Marcel (2010), Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry, Springer-Verlag, pp. 295–296, ISBN 9783540709978.
  2. ^ Gage, M. E. (1984), "Curve shortening makes convex curves circular", Inventiones Mathematicae, 76 (2): 357–364, doi:10.1007/BF01388602, MR 0742856.
  3. ^ Chow, Bennett; Knopf, Dan (2004), The Ricci Flow: An Introduction, Mathematical surveys and monographs, vol. 110, American Mathematical Society, p. 157, ISBN 9780821835159.
  4. ^ Grady, Leo J.; Polimeni, Jonathan (2010), Discrete Calculus: Applied Analysis on Graphs for Computational Science, Springer-Verlag, p. 275, ISBN 9781849962902.