In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov,[1] and later independently by Christer Borell,[2] states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian boundary measure.
endowed with the standard Gaussian measure
Then the Gaussian isoperimetric inequality states that where The original proofs by Sudakov, Tsirelson and Borell were based on Paul Lévy's spherical isoperimetric inequality.
Sergey Bobkov proved Bobkov's inequality, a functional generalization of the Gaussian isoperimetric inequality, proved from a certain "two point analytic inequality".
[3] Bakry and Ledoux gave another proof of Bobkov's functional inequality based on the semigroup techniques which works in a much more abstract setting.
[4] Later Barthe and Maurey gave yet another proof using the Brownian motion.