Isoperimetric Problems


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Isoperimetric Problems

 

a class of problems of the calculus of variations. The simplest isoperimetric problems (for example, finding triangles and polygons of given perimeter that have the greatest area; finding the closed curve of given length that bounds the maximum area; determining the closed surface of given area and maximum volume) were known to ancient Greek scholars (Archimedes, Zenodorus, and others).

The general study of isoperimetric problems was begun in 1697, when Jakob Bernoulli published an isoperimetric problem that he posed and partially solved: of all curves of given length, find the curve for which some quantity dependent on the curve reaches a minimum or maximum. A systematic study of isoperimetric problems was first made in 1732 by L. Euler. An example of an isoperimetric problem is: of curves of given length l passing through the points A and B, find the curve for which the area

Figure 1

of a curvilinear trapezoid (shaded in Figure 1) is greatest. The area of a curvilinear trapezoid is equal to

and the length of the arc is

Consequently, the problem reduces to finding the largest value of integral (1) under conditions (2). It turns out that the unknown curve is an arc of a circle.

REFERENCE

Lavrent’ev, M. A., and L.A. Moscow-Leningrad, 1950.
References in periodicals archive ?
They used this interpretation to relate the Tanny sequence with the so called discrete connected isoperimetric problem on infinite complete binary trees.
The discrete isoperimetric problem on G is to find the value of [b.
1 is used in [2] to relate the Tanny value to connected discrete isoperimetric problem on [T.
The following theorem is from [2] which relates the connected isoperimetric problem on [T.
Sunil Chandran, Anita Das, Isoperimetric Problem and Meta-Fibonacci Sequences, In the Proceeding of 14th Annual International Computing and Combinatorics Conference 2008, Dalian, China.