Isoperimeterprofil

Isoperimeterprofil, also known as the isoperimetric profile, is a function in differential geometry that describes how small the boundary of a region must be to enclose a given volume in a geometric space. For a Riemannian manifold M with volume measure Vol and boundary area Area, the isoperimeterprofil I_M assigns to each volume v in (0, Vol(M)) the infimum of the boundary areas of all regions with that volume:

I_M(v) = inf{ Area(∂Ω) : Ω ⊂ M, Vol(Ω) = v }.

If M is compact, the domain is (0, Vol(M)). The infimum is taken over regions with smooth boundary or finite perimeter, and ∂Ω is interpreted in the appropriate geometric sense.

Basic properties: I_M is nondecreasing in v; under mild regularity it is continuous and often locally Lipschitz. In many settings it is differentiable almost everywhere, and its derivative is related to the mean curvature of minimal boundary hypersurfaces achieving or approximating the infimum.

Examples: In Euclidean space R^n, the classical isoperimetric inequality yields I_{R^n}(v) = n ω_n^{1/n} v^{(n-1)/n}, with equality for balls. On spaces of constant curvature, minimizers are spherical caps or their analogues, and the profile can be expressed in terms of the geometry of those spaces.

Applications: the isoperimetric profile encodes geometric and analytic information about M, relates to isoperimetric and Sobolev inequalities, and connects with invariants such as the Cheeger constant. It is used in comparison geometry and geometric analysis to study volume growth, minimal surfaces, and diffusion phenomena.