Isoperimetric problem

The isoperimetric problem of the geometric calculus of variations asks in its original form, which goes back to classical Greece (see problem of Dido ), which form a closed curve of a given length must have so that this curve spans the largest area . ${\ displaystyle \, l}$ ${\ displaystyle \, F}$

The term is also applied to various generalizations of the question.

The classic isoperimetric problem

The name isoperimetric means in Greek of the same extent . Even the Greeks knew that the solution to the (classical) isoperimetric problem is the circle, and that this is a consequence of the isoperimetric inequality

{\ displaystyle 4 \ pi F \ leq l ^ {2}}

is: The equality holds only for the circle (this is true and , the circle radius). ${\ displaystyle l = 2 \ pi r}$ ${\ displaystyle F = \ pi r ^ {2}}$ ${\ displaystyle r}$

The oldest, incomplete, attempt at proof was made by Zenodorus in the 2nd century BC. Undertaken. Complete evidence for the vividly very plausible fact was only produced in the 19th century. In 1838 Jakob Steiner gave a purely geometric proof that the solution (if it exists) must be a convex, symmetrical curve. Only F. Edler (1882) for the plain and Karl Weierstraß and Hermann Amandus Schwarz (1884) for the room gave complete evidence . In 1902, Adolf Hurwitz gave a simple proof of piecewise continuous boundary curves using Fourier series. Further evidence comes from Erhard Schmidt (1938) , for example .

There are also higher dimensional generalizations of the isoperimetric problem. For example, the sphere has the smallest surface area in three dimensions of all surfaces that span a given volume. This can be seen clearly from the spherical shape of soap bubbles, which try to make their surface tension and thus their surface area as small as possible. This was first proved mathematically by Hermann Amandus Schwarz in 1884. The cases of the sphere in more than three dimensions were proven by Edgar Krahn in 1925 and for non-Euclidean geometries by Erhard Schmidt.

Related problems in mathematical physics are also referred to as isoperimetric problems, for example the conjecture by Barré de Saint-Venant (1856) that elastic rods with a circular cross-section have maximum torsional stiffness .

The isoperimetric problem of the calculus of variations

In the calculus of variations one speaks more generally of the following problem of an isoperimetric problem:

Let there be. We are looking for a function for which the functional ${\ displaystyle \ alpha, \ beta, l \ in \ mathbb {R}}$ ${\ displaystyle F}$

{\ displaystyle I (u) = \ int \ limits _ {a} ^ {b} F (x, u (x), u '(x)) {\ rm {d}} x}

among all functions that and as well ${\ displaystyle u (x)}$ ${\ displaystyle \, u (a) = \ alpha}$ ${\ displaystyle \, u (b) = \ beta}$

{\ displaystyle K (u) = \ int \ limits _ {a} ^ {b} G (x, u (x), u '(x)) {\ rm {d}} x = l}

meet, becomes extremal. In the special case , this boundary condition requires that the scope of a curve described by is constant. ${\ displaystyle G (x, u (x), u '(x)) = {\ sqrt {1 + u' (x) ^ {2}}}}$ ${\ displaystyle \, l}$ ${\ displaystyle u (x)}$

The solution to the problem arises with the Lagrange function

{\ displaystyle L = F + \ lambda \, G}

from the Euler equation

{\ displaystyle {\ frac {\ partial L} {\ partial u}} - {\ frac {d} {dx}} \, {\ frac {\ partial L} {\ partial u '}} = 0.}

Evidence sketch of the classic problem for the plane case

We are following Jakob Steiner's proof mentioned above.

Steiner's argument for the convexity of the area

Steiner treated the problem in two and three dimensions and assumed the existence of a solution. In two dimensions, he first showed that the surface he was looking for is a convex set (that is, every line that connects two edge points lies entirely within the surface). If this were not the case, you would have a situation like the one in the illustration on the right: You could mirror the curve on the connecting straight line and thus get a larger area with the same circumference. The maximum area sought must therefore be convex.

The Steiner problem can further be reduced to the fact that one considers convex surfaces that are bounded by a segment AB and curves of fixed length between points A and B. Because every segment AB that divides the perimeter of the maximum area sought also divides the area. If this were not the case and if, for example, the sub-area below the segment AB were larger, the smaller area above the straight line could be replaced by the area below the segment reflected at AB and thus an area with larger content with the same circumference would be obtained. The problem is thus reduced to finding a convex curve of a given circumference with the end points A, B on a straight line AB, so that the area between the curve and AB is maximal.

In a last step Steiner then proves that of all convex curves over the base segment AB with the same circumference, the semicircle has the largest content. For at any point C on the curve, consider the triangle ACB. The area F between the curve and the segment AB is divided into the area F3 of the triangle ACB and the areas F1 between the curve and the triangle side AC and F2 between the curve and the side CB. Now vary the triangle ACB by shifting B on the straight line AB, but the distances AC, CB remain the same. Of all these triangles, the triangle with the right angle in C has the largest area F3. If the angle in ABC at point C is not a right angle, the curve can be replaced by one of the same circumference, the area F being composed of the area of the right triangle and the areas F1 and F2 over the triangle sides AC, CB, their Length was unchanged. The curve we are looking for has right angles at any point C on the curve and is therefore a semicircle according to Thales' theorem .

literature

Richard Courant , Harold Robbins : What is Mathematics? Springer 1973, p. 283 (brief explanation of Steiner's proof).
Helmuth Gericke : On the history of the isoperimetric problem. In: Mathematical semester reports. To maintain the connection between school and university. Volume XXIX (1982), ISSN 0720-728X , pp. 160-187 (with evidence from Zenodorus and sketch of a variant of Steiner's evidence).
Peter Gruber On the history of convex geometry and the geometry of numbers. In: Hirzebruch u. a .: A Century of Mathematics 1890–1990. Vieweg 1990 (history).
Hugo Hadwiger : Lectures on content, surface and isoperimetry. Springer 1957.
Robert Osserman : The isoperimetric inequality. Bulletin AMS, 84, 1978, pp. 1182-1238, online .
Burago , Zalgaller : Geometric inequalities. Springer 1988.
G. Talenti: The standard isoperimetric problem. In: Gruber, Wills: Handbook of Convex Geometry. North Holland 1993, pp. 73-123.
Wilhelm Blaschke : circle and sphere. 2nd edition, De Gruyter 1956.
Isaac Chavel: Isoperimetric Inequalities. Cambridge University Press 2001.

Web links

Uni-Magazin Universität Halle on the isoperimetric problem. ( Memento from January 3, 2015 in the web archive archive.today ).
Hopf, Samelson: Selected chapters in geometry. With a chapter on the isoperimetric problem (PDF; 221 kB).
Viktor Blasjö: Evolution of the isoperimetric problem. American Mathematical Monthly, Volume 112, 2005, p. 526.
Wiegert: The sagacity of circles. A history of the isoperimetric problem. MAA

Individual evidence

↑ Anton Nokk ( arrangement ): Zenodorus' treatise on the isoperimetric figures, based on the excerpts that the Alexandrians Theon and Pappus have given us from the same, edited in German by Dr. Nokk. In: Program of the Grand Ducal Lyceum in Freiburg im Breisgau - as an invitation to public exams. 1860, supplement, pp. 1-35. Digitized , resource of the Bavarian State Library .

↑ Peter Gustav Lejeune Dirichlet was the first to point out the question of existence .

↑ Hurwitz: Quelques applications geometriques des series de Fourier. Annales de l'Ecole Normale, Vol. 19, 1902, pp. 357-408. The proof can be found, for example, in Blaschke: Lectures on differential geometry. Vol. 1, Springer, 1924, p. 45.

↑ Dissertation with Richard Courant in Göttingen 1925: On the minimal properties of the sphere in three and more dimensions.

^ Kurt Meyberg, Peter Vachenauer: Höhere Mathematik 2. Springer Verlag, 4th edition 2001, p. 428 f.

^ John Clegg calculus of variations. Teubner, 1970, p. 87.

↑ The area is the product of the base length AC and the height, which is only CB for a right angle in C and is smaller otherwise.

[1] Anton Nokk ( arrangement ): Zenodorus' treatise on the isoperimetric figures, based on the excerpts that the Alexandrians Theon and Pappus have given us from the same, edited in German by Dr. Nokk. In: Program of the Grand Ducal Lyceum in Freiburg im Breisgau - as an invitation to public exams. 1860, supplement, pp. 1-35. Digitized , resource of the Bavarian State Library .

[2] Peter Gustav Lejeune Dirichlet was the first to point out the question of existence .

[3] Hurwitz: Quelques applications geometriques des series de Fourier. Annales de l'Ecole Normale, Vol. 19, 1902, pp. 357-408. The proof can be found, for example, in Blaschke: Lectures on differential geometry. Vol. 1, Springer, 1924, p. 45.

[4] Dissertation with Richard Courant in Göttingen 1925: On the minimal properties of the sphere in three and more dimensions.

[5] Kurt Meyberg, Peter Vachenauer: Höhere Mathematik 2. Springer Verlag, 4th edition 2001, p. 428 f.

[6] John Clegg calculus of variations. Teubner, 1970, p. 87.

[7] The area is the product of the base length AC and the height, which is only CB for a right angle in C and is smaller otherwise.