isoperimetrischen

Isoperimetrisch is an adjective used in German mathematics to describe properties and problems related to isoperimetry, the study of figures that minimize boundary length for a given area or, equivalently, maximize area for a given boundary. The isoperimetric problem is one of the oldest in geometry and in the calculus of variations.

In the Euclidean plane, the isoperimetric inequality states that among all simple closed curves of length L, the enclosed area A satisfies A ≤ L^2/(4π). Equality holds only for a circle, making the circle the isoperimetric minimizer for a given perimeter.

In higher dimensions, the analogous statement says that among bodies with a given surface area, the volume is maximized by balls; equivalently, for a fixed volume, balls minimize surface area. These inequalities extend to Euclidean space of any dimension and to more general spaces, where the isoperimetric profile encodes the best possible boundary size for each volume.

Discrete versions appear in graph theory, where the edge boundary of a vertex set is bounded below by a function of its size; these isoperimetric inequalities relate to expansion properties and spectral gaps (Cheeger constants).

Applications arise in physics and materials science, in particular the behavior of soap films, foams, and cell shapes, where energy minimization leads to boundary-minimizing configurations.

In German mathematical writing, isoperimetrisch is the common attributive form related to Isoperimetrie, used in phrases such as isoperimetrische Ungleichung.