the Gaussian isoperimetric inequality
Among all sets of a given Gaussian measure, which one has the smallest 'boundary' — equivalently, the slowest-growing enlargement? The classical isoperimetric inequality says balls minimize surface area for fixed volume; the Gaussian isoperimetric inequality (Borell; Sudakov-Tsirelson) says that for the standard Gaussian measure the extremal sets are HALF-SPACES. This single geometric fact is the deepest source of Gaussian concentration of measure — the phenomenon that any not-too-wild function of many Gaussians is nearly constant.
Let gamma_n be the standard Gaussian measure on R^n and write A_r = { x : dist(x, A) <= r } for the r-enlargement of a set A. The inequality states: among all measurable A with a given gamma_n(A), the enlargement gamma_n(A_r) is minimized when A is a half-space. Writing Phi for the standard normal CDF and Phi^(-1) for its inverse, the sharp statement is gamma_n(A_r) >= Phi( Phi^(-1)(gamma_n(A)) + r ) for all r >= 0, with equality for half-spaces. The dimension n does NOT appear in the bound — the right-hand side is one-dimensional. The immediate consequence is concentration: if gamma_n(A) >= 1/2, then gamma_n(A_r) >= Phi(r) >= 1 - exp(-r^2/2), so the r-neighborhood of any half-mass set captures essentially all the measure once r is a few units. Equivalently, every 1-Lipschitz function f satisfies gamma_n( | f - median f | >= r ) <= 2 exp(-r^2/2): Lipschitz functions of a standard Gaussian vector are sub-Gaussian around their median with a dimension-free constant.
Why it matters: this is the engine behind Borell-TIS (the supremum of a Gaussian process is a Lipschitz function of the underlying Gaussian vector, hence concentrated), behind the dimension-free behaviour of high-dimensional Gaussian models, and behind the log-Sobolev / hypercontractivity story (the Gaussian isoperimetric profile and the log-Sobolev inequality are two facets of the same Gaussian curvature). A common slip to avoid: the constant is genuinely dimension-free, but the function must be Lipschitz with respect to the EUCLIDEAN distance scaled by the Gaussian's standard deviation; a function that is steep in some directions need not concentrate.
The Euclidean norm f(x) = ||x|| on R^n is 1-Lipschitz. Gaussian isoperimetry gives gamma_n( | ||x|| - m | >= r ) <= 2 exp(-r^2/2) where m is the median of ||x|| (which is about sqrt(n)). So a standard Gaussian vector in high dimensions lives in a thin shell of radius about sqrt(n) and width O(1) — concentration of the norm, free from the isoperimetric inequality.
Half-spaces are extremal: every 1-Lipschitz function of a Gaussian concentrates with a dimension-free Gaussian tail.
The concentration constant is dimension-free, but Lipschitz is measured in the Euclidean (sigma-scaled) metric; a function steep in some coordinate can still fluctuate a lot.