Isoperimetric ratio

In analytic geometry, the isoperimetric ratio of a simple closed curve in the Euclidean plane is the ratio {{math|L²/A}}, where {{mvar|L}} is the length of the curve and {{mvar|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, 4{{pi}}, for a circle; any other curve has a larger value.{{citation|title=Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry|first=Marcel|last=Berger|publisher=Springer-Verlag|year=2010|isbn=9783540709978|pages=295–296|url=https://books.google.com/books?id=pN0iAVavPR8C&pg=PA295}}. 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{{pi}}.{{citation

| last = Gage | first = M. E. | authorlink = Michael Gage (mathematician)

| doi = 10.1007/BF01388602

| issue = 2

| journal = Inventiones Mathematicae

| mr = 742856

| pages = 357–364

| title = Curve shortening makes convex curves circular

| volume = 76

| year = 1984}}.

For higher-dimensional bodies of dimension d, the isoperimetric ratio can similarly be defined as {{math|B^d/V^{d − 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).{{citation|title=The Ricci Flow: An Introduction|series=Mathematical surveys and monographs|volume=110|publisher=American Mathematical Society|first1=Bennett|last1=Chow|first2=Dan|last2=Knopf|year=2004|isbn=9780821835159|page=157|url=https://books.google.com/books?id=BGU_msH91EoC&pg=PA157}}. Other related quantities include the Cheeger constant of a Riemannian manifold and the (differently defined) Cheeger constant of a graph.{{citation|title=Discrete Calculus: Applied Analysis on Graphs for Computational Science|first1=Leo J.|last1=Grady|first2=Jonathan|last2=Polimeni|publisher=Springer-Verlag|year=2010|isbn=9781849962902|page=275|url=https://books.google.com/books?id=E3-OSVSPbU0C&pg=PA275}}.