The theorem of the 4 tops consititue a remarkable result of differential Geometry as for the total properties of the closed curved .

Statement

Statement in the simplest case

That is to say $\backslash \; gamma$ a convex closed curve , parameterized by its Length S . That is to say K (S) the Curve calculated at the point $\backslash \; gamma$ (S) . Then there exists at least 4 parameters $s\_i$ for which $k\text{'}\; (s\_i)\; =\; 0$.

The geometrical significance of this result is that the curve is either constant, or has at least 4 extrema. One will be able to find of it a demonstration in [ManDoCar]

General case

The theorem of the 4 tops with initially shown for the convex curves (i.e. with positive curve) in 1909 by Syamadas Mukhopadhyaya. Its proof uses makes it that a point of the curve is a extremum function curve If and only if the osculatory Cercle in this point has a contact in 4 points with the curve (in the general case, the osculatory circle has 3 contact points with the curve)). The theorem of the 4 tops was shown in the general case by Adolf Kneser in 1912.

Reciprocal

The reciprocal one of the theorem of the four tops states that a function continues, with real value , and having for Domaine of definition the circle unit and who in addition has 2 maximum local and 2 minimum is the curve of a curve simple and closed plan. The reciprocal one was proven for strictly positive functions in 1971 by Hermann Gluck, as a particular case of a more general theorem concerning the precalculation da the curve on a N-spheres. The reciprocal one was finally shown in the general case by Björn Dahlberg little time before its in January 1998 death and was published in posthumous title. The proof of Dahlberg uses mainly the index, argument which one in addition finds in certain demonstration of the fundamental theorem of the algebra.

Sources