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:
“A man willing to work, and unable to find work, is perhaps the saddest sight that fortune’s inequality exhibits under this sun.”
—Thomas Carlyle (1795–1881)
“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)
“Is it true or false that Belfast is north of London? That the galaxy is the shape of a fried egg? That Beethoven was a drunkard? That Wellington won the battle of Waterloo? There are various degrees and dimensions of success in making statements: the statements fit the facts always more or less loosely, in different ways on different occasions for different intents and purposes.”
—J.L. (John Langshaw)