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).
Read more about this topic: Isoperimetric Inequality
Famous quotes containing the words inequality, higher and/or dimensions:
“Nature is unfair? So much the better, inequality is the only bearable thing, the monotony of equality can only lead us to boredom.”
—Francis Picabia (1878–1953)
“By the artist’s seizing any one object from nature, that object no longer is part of nature. One can go so far as to say that the artist creates the object in that very moment by emphasizing its significant, characteristic, and interesting aspects or, rather, by adding the higher values.”
—Johann Wolfgang Von Goethe (1749–1832)
“Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
—bell hooks (b. c. 1955)