Four-vertex theorem

The Four-vertex theorem states that the curvature function of a simple, closed plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex.

The Four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. [cite journal | last=Mukhopadhyaya | first=S. | title=New methods in the geometry of a plane arc | journal=Bull. Calcutta Math. Soc.| year=1909 | volume=1 | pages=21–27] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has 4-point contact with the curve (in general the osculating circle has only 3-point contact with the curve). The Four-vertex theorem was proved in general by Adolf Kneser in 1912 using a projective argument. [cite conference | last=Kneser | first=Adolf | title=Bemerkungen uber die Anzahl der Extrema des Krummung auf geschlossenen Kurven und uber verwandte Fragen in einer nicht eucklidischen Geometrie | booktitle=Festschrift Heinrich Weber | publisher=Teubner | year=1912 | pages=170-180]

The converse to the Four-vertex theorem states that any continuous, real-valued function of the circle that has at least two local maxima and two local minima is the curvature function of a simple, closed plane curve. The converse was proved for strictly positive functions in 1971 by Herman Gluck as a special case of a general theorem on pre-assigning the curvature of n-spheres. [cite journal | last=Gluck | first=Herman | title=The converse to the four-vertex theorem | journal=L'Enseignement Math. | volume=17 | pages=295–309 | year=1971] The full converse to the Four-vertex theorem was proved by Björn Dahlberg shortly before his death in January, 1998 and published posthumously. [cite journal | last=Dahlberg | first=Björn | title=The converse of the four vertex theorem | journal=Proc. Amer. Math. Soc. | volume=133 | issue=7 | pages=2131–2135 | year=2005 | url=http://www.ams.org/proc/2005-133-07/S0002-9939-05-07788-9/home.html | doi=10.1090/S0002-9939-05-07788-9 ] Dahlberg's proof uses a winding number argument which is in some ways reminiscent of the standard topological proof of the Fundamental Theorem of Algebra. [cite journal|title=The Four Vertex Theorem and Its Converse|journal=Notices of the American Mathematical Society|volume=54|issue=2|year=2007|author=DeTruck, D., Gluck, H., Pomerleano, D., and Vick, D.S.]

One corollary of the theorem is that a homogeneous, planar disk rollingon a horizontal surface under gravity has at least 4 balance points.The 3D generalization is not trivial, in fact, one can show that there exist convex, homogeneous objects with less than 4 balance points,see Gomboc.

Notes

External links

* [http://www.ams.org/notices/200702/fea-gluck.pdf The Four Vertex Theorem and Its Converse] -- An expository article which explains Robert Osserman's simple proof of the Four-vertex theorem and Dahlberg's proof of its converse, offers a brief overview of extensions and generalizations, and gives biographical sketches of Mukhopadhyaya, Kneser and Dahlberg.