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:
“However energetically society in general may strive to make all the citizens equal and alike, the personal pride of each individual will always make him try to escape from the common level, and he will form some inequality somewhere to his own profit.”
—Alexis de Tocqueville (1805–1859)
“For human nature, being more highly pitched, selved, and distinctive than anything in the world, can have been developed, evolved, condensed, from the vastness of the world not anyhow or by the working of common powers but only by one of finer or higher pitch and determination than itself.”
—Gerard Manley Hopkins (1844–1889)
“It seems to me that we do not know nearly enough about ourselves; that we do not often enough wonder if our lives, or some events and times in our lives, may not be analogues or metaphors or echoes of evolvements and happenings going on in other people?—or animals?—even forests or oceans or rocks?—in this world of ours or, even, in worlds or dimensions elsewhere.”
—Doris Lessing (b. 1919)