Isoperimetric Inequality in Higher Dimensions
The isoperimetric theorem generalizes to surfaces in the three-dimensional Euclidean space. Among all simple closed surfaces with given surface area, the sphere encloses a region of maximal volume. An analogous statement holds in Euclidean spaces of any dimension.
In full generality (Federer 1969, §3.2.43), the isoperimetric inequality states that for any set S ⊂ Rn whose closure has finite Lebesgue measure
where M*n-1 is the (n-1)-dimensional Minkowski content, Ln is the n-dimensional Lebesgue measure, and ωn is the volume of the unit ball in Rn. If the boundary of S is rectifiable, then the Minkowski content is the (n-1)-dimensional Hausdorff measure.
The isoperimetric inequality in n-dimensions can be quickly proven by the Brunn-Minkowski inequality (Osserman (1978); Federer (1969, §3.2.43)).
The n-dimensional isoperimetric inequality is equivalent (for sufficiently smooth domains) to the Sobolev inequality on Rn with optimal constant:
for all u ∈ W1,1(Rn).
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