# Gaussian isoperimetric inequality
The **Gaussian isoperimetric inequality** is a theorem of probability and geometric analysis stating that, among all Borel subsets of \( \mathbb{R}^n \) with a prescribed [Gaussian measure](https://www.edgepedia.org/gaussian-measure), half-spaces have the smallest Gaussian perimeter. It was proven in the 1970s by V. N. Sudakov and B. S. Tsirelson, and independently by [Christer Borell](https://www.edgepedia.org/christer-borell)[1](https://arxiv.org/abs/1202.4124) • [2](https://cir.nii.ac.jp/crid/1360011146621655040). It is the Gaussian analog of the classical isoperimetric inequality, in which the ball minimizes perimeter for a given volume; in Gauss space the minimizer is a half-space instead[3](https://link.springer.com/article/10.1007/s13373-015-0074-x).
| Key fact | Detail |
|---|---|
| Statement | For a Borel set \( A \subset \mathbb{R}^n \) with \( \gamma(A) = a \), the Gaussian perimeter satisfies \( \gamma^+(\partial A) \ge \frac{1}{\sqrt{2\pi}} e^{-\Phi^{-1}(a)^2/2} \), where \( \Phi \) is the standard normal distribution function[4](https://glivshyts6.math.gatech.edu/Conc-lecture-notes.pdf) |
| Extremal sets | Half-spaces \( H_a = \{x : x_1 \le \Phi^{-1}(a)\} \) attain equality; Carlen and Kerce later characterized them as the unique minimizers[4](https://glivshyts6.math.gatech.edu/Conc-lecture-notes.pdf) • [3](https://link.springer.com/article/10.1007/s13373-015-0074-x) |
| Dimension independence | The isoperimetric profile \( \mathcal{I}_\gamma \) is the same in every dimension, coinciding with the one-dimensional profile[5](https://onlinelibrary.wiley.com/doi/10.1002/cpa.70020) |
| Integrated form | If \( \gamma(A) = \Phi(a) \), then \( \gamma(A_r) \ge \Phi(a+r) \) for every \( r \ge 0 \), where \( A_r \) is the Euclidean \( r \)-neighborhood of \( A \)[6](https://www.math.univ-toulouse.fr/~ledoux/LevyGromov.pdf) |
| History | Sudakov–Tsirelson (1974, English translation 1978) and Borell (1975) proved it independently[7](https://cris.tau.ac.il/en/publications/extremal-properties-of-half-spaces-for-spherically-invariant-meas/) • [2](https://cir.nii.ac.jp/crid/1360011146621655040) |
| Main consequence | Gaussian concentration: Lipschitz functionals of a Gaussian vector concentrate at least as strongly as a one-dimensional normal law[8](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MV/MVspa/Lifshits%20M.%20Lectures%20on%20Gaussian%20processes%20%28Springer%2C%202012%29%28ISBN%209783642249389%29%28O%29%28133s%29_MVspa_.pdf) |
## Setting and statement
The Gauss space is \( \mathbb{R}^n \) endowed with the standard Gaussian measure
\[ \gamma(E) = \frac{1}{(2\pi)^{n/2}} \int_E e^{-|x|^2/2}\, dx, \]
a probability measure that is rotation invariant but not translation invariant[3](https://link.springer.com/article/10.1007/s13373-015-0074-x). For a measurable set \( E \), the Gaussian perimeter \( P_\gamma(E) \), equivalently the boundary measure \( \gamma^+(\partial A) \), is defined as a Minkowski content, the lower limit of \( (\gamma(A_r) - \gamma(A))/r \) as \( r \downarrow 0 \)[4](https://glivshyts6.math.gatech.edu/Conc-lecture-notes.pdf) • [9](https://androma.org/theorems/6759).
The inequality states that if \( A \) is Borel measurable in \( \mathbb{R}^n \) with \( \gamma(A) = a \in [0,1] \), then
\[ \gamma^+(\partial A) \ge \gamma^+(\partial H_a) = \frac{1}{\sqrt{2\pi}} e^{-\Phi^{-1}(a)^2/2}, \]
where \( H_a = \{x \in \mathbb{R}^n : x_1 \le \Phi^{-1}(a)\} \) is the half-space of Gaussian measure \( a \)[4](https://glivshyts6.math.gatech.edu/Conc-lecture-notes.pdf). The right-hand side defines the Gaussian isoperimetric function
\[ I_\gamma(a) = \frac{1}{\sqrt{2\pi}} e^{-\Phi^{-1}(a)^2/2}, \]
which satisfies the identities \( I_\gamma(a) = 1/(\Phi^{-1}(a))' \) and \( I_\gamma(a) \cdot I_\gamma''(a) = -1 \)[4](https://glivshyts6.math.gatech.edu/Conc-lecture-notes.pdf). In the equivalent analytic form, if \( \gamma(E) = \Phi(s) \) then \( P_\gamma(E) \ge e^{-s^2/2} \), with equality if and only if \( E \) is a half-space \( H_{\nu,s} \) for some \( \nu \in \mathbb{S}^{n-1} \)[3](https://link.springer.com/article/10.1007/s13373-015-0074-x).
A striking feature is that the bound does not depend on the dimension \( n \). Sudakov and Tsirelson showed that half-planes are isoperimetric minimizers in \( (\mathbb{R}^{n-1}, \gamma^{n-1}) \), and the product structure of the Gaussian measure then implies that the isoperimetric profile \( \mathcal{I}_{\gamma^{n-1}} \) coincides with the one-dimensional profile \( \mathcal{I}_\gamma \)[5](https://onlinelibrary.wiley.com/doi/10.1002/cpa.70020). The inequality also extends to infinite-dimensional settings: for a Gaussian measure on a separable [Banach space](https://www.edgepedia.org/banach-space), \( \Phi^{-1}(P^*(A_r)) \ge \Phi^{-1}(P(A)) + r \), with equality attained on half-spaces[8](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MV/MVspa/Lifshits%20M.%20Lectures%20on%20Gaussian%20processes%20%28Springer%2C%202012%29%28ISBN%209783642249389%29%28O%29%28133s%29_MVspa_.pdf).
## History and proofs
The theorem was established independently by Sudakov and Tsirelson in 1974, published in Russian in *Zapiski LOMI* and translated into English in the *Journal of Soviet Mathematics* in 1978 under the title "Extremal properties of half-spaces for spherically invariant measures", and by Christer Borell, then at Uppsala, in "The Brunn-Minkowski inequality in Gauss space", *Inventiones Mathematicae* 30 (1975), pp. 207–216[7](https://cris.tau.ac.il/en/publications/extremal-properties-of-half-spaces-for-spherically-invariant-meas/) • [2](https://cir.nii.ac.jp/crid/1360011146621655040). Borell's proof passes to the limit from the classical isoperimetric inequality on high-dimensional spheres, following Poincaré's observation that [spherical measures](https://www.edgepedia.org/spherical-measure) converge to Gaussian distributions[10](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2023/06/Borell-Subspace-original.pdf).
**Ehrhard's inequality.** In the early 1980s A. Ehrhard gave a different proof using a Gaussian symmetrization procedure analogous to Steiner symmetrization, and with the same tool established a Brunn–Minkowski type inequality stronger than the log-concavity of Gaussian measures, valid when at least one of the two sets is convex[11](https://duch.mimuw.edu.pl/~rlatala/papers/gaussicm02.pdf). In its common form, for Borel sets \( A, B \subset \mathbb{R}^n \) and \( t \in [0,1] \),
\[ \Phi^{-1}\big(\gamma_n((1-t)A + tB)\big) \ge (1-t)\,\Phi^{-1}(\gamma_n(A)) + t\,\Phi^{-1}(\gamma_n(B)). \]
The Gaussian isoperimetric inequality follows by choosing \( B = rB_2^n \) and letting \( t \to 0 \)[12](https://perso.ens-lyon.fr/aurelien.garivier/www.math.univ-toulouse.fr/_agarivie/concentration/AGG-exo-2.pdf). Equality holds when \( A \) and \( B \) are half-spaces with one contained in the other[12](https://perso.ens-lyon.fr/aurelien.garivier/www.math.univ-toulouse.fr/_agarivie/concentration/AGG-exo-2.pdf). Symmetrization proofs show that the Gaussian perimeter strictly decreases under Ehrhard symmetrization of a set in a given direction, unless the one-dimensional sections parallel to that direction are half-lines or lines[13](https://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub97.pdf).
**Functional and semigroup proofs.** S. G. Bobkov, of Syktyvkar University, gave in 1997 an elementary proof via an isoperimetric inequality on the discrete cube, the precise analog of a logarithmic inequality due to Talagrand; the Gaussian result is derived through a two-point inequality and the central limit theorem[14](https://www-users.cse.umn.edu/~bobkov/papers/1997_AOP.pdf) • [6](https://www.math.univ-toulouse.fr/~ledoux/LevyGromov.pdf). His functional form of the inequality, for smooth \( f : \mathbb{R}^k \to [0,1] \), reads
\[ U\!\left(\int f\, d\gamma\right) \le \int \sqrt{U(f)^2 + |\nabla f|^2}\, d\gamma, \]
with \( U = \varphi \circ \Phi^{-1} \) and \( \varphi = \Phi' \)[6](https://www.math.univ-toulouse.fr/~ledoux/LevyGromov.pdf). In 1996, Dominique Bakry and Michel Ledoux, of Université Paul–Sabatier in Toulouse, established a Lévy–Gromov isoperimetric inequality for the invariant measure of an infinite-dimensional diffusion generator of positive curvature, with the Gaussian measure as the isoperimetric model; their elementary semigroup arguments provide another proof of Gaussian isoperimetry, and the resulting inequality strengthens the classical logarithmic [Sobolev inequality](https://www.edgepedia.org/sobolev-inequality) in this context[6](https://www.math.univ-toulouse.fr/~ledoux/LevyGromov.pdf).
**Equality cases.** The characterization of half-spaces as the unique minimizers is a relatively recent result of Carlen and Kerce, relying on the theory of rearrangements and on probabilistic techniques involving the Mehler semigroup of contractions; a different proof uses Ehrhard symmetrization in the role that Steiner symmetrization plays in De Giorgi's proof of the Euclidean isoperimetric inequality[3](https://link.springer.com/article/10.1007/s13373-015-0074-x).
## Consequences and applications
**Gaussian concentration.** The concentration principle is a direct corollary: the distribution of a Lipschitz functional of a Gaussian vector is at least as concentrated as that of a univariate normal law[8](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MV/MVspa/Lifshits%20M.%20Lectures%20on%20Gaussian%20processes%20%28Springer%2C%202012%29%28ISBN%209783642249389%29%28O%29%28133s%29_MVspa_.pdf). Concretely, if \( X \) is a centered Gaussian random vector in a separable Banach space \( (F, \|\cdot\|) \), then for any \( t > 0 \)
\[ P\big(|\|X\| - \mathrm{Med}(\|X\|)| \ge t\big) \le 2\big(1 - \Phi(t/\sigma)\big) \le e^{-t^2/2\sigma^2}, \]
where \( \sigma = \sup\{\sqrt{E(x^*(X))^2} : x^* \in F^*,\ \|x^*\| \le 1\} \)[11](https://duch.mimuw.edu.pl/~rlatala/papers/gaussicm02.pdf). The inequality is a standard tool in high-dimensional probability, where it yields concentration for Lipschitz functions[15](https://webapps.math.uci.edu/~rvershyn/teaching/hdp/hdp.html), and it extends to infinite products: for i.i.d. standard normals \( X = (X_1, X_2, \ldots) \) with distribution \( \gamma_N \), \( \Phi^{-1}(\gamma_N(A_\varepsilon)) \ge \Phi^{-1}(\gamma_N(A)) + \varepsilon \) for every measurable \( A \subseteq \mathbb{R}^N \)[16](https://math.iisc.ac.in/~manju/GP/6-Concentration%20and%20comparison%20again.pdf).
**Stability.** Quantitative versions control how close a near-minimizing set must be to a half-space. If \( A \subset \mathbb{R}^n \) satisfies \( \gamma_n(A) = 1/2 \) and \( \gamma_n^+(A) \le (2\pi)^{-1/2} + \delta \), then there exists a half-space \( B \) with \( \gamma_n(A \Delta B) \le C \log^{-1/2}(1/\delta) \) for an absolute constant \( C \); an earlier result of Cianchi and coauthors gave \( \gamma_n(A \Delta B) \le C(n)\sqrt{\delta} \) with no effective bound on \( C(n) \)[1](https://arxiv.org/abs/1202.4124). The dimension independence of the inequality itself suggests that constants in quantitative versions should also be independent of \( n \), a property with applications in probability[3](https://link.springer.com/article/10.1007/s13373-015-0074-x).
**Extensions.** The inequality has been generalized in several directions. Bakry and Ledoux's framework extends it to the Wiener measure on the paths of a [Riemannian manifold](https://www.edgepedia.org/riemannian-manifold) with bounded [Ricci curvature](https://www.edgepedia.org/ricci-curvature)[6](https://www.math.univ-toulouse.fr/~ledoux/LevyGromov.pdf). An anisotropic version, for Gaussian measures weighted by a matrix \( A \), has been proven with its equality cases characterized; there Ehrhard symmetrization can fail to decrease the anisotropic Gaussian perimeter, and a corrected inequality with an error term was found[17](https://arxiv.org/abs/2309.03853). The sharp Gaussian isoperimetric inequality has also been proven for conjugate heat-kernel measures along a [Ricci flow](https://www.edgepedia.org/ricci-flow), via a monotonicity formula[18](https://arxiv.org/abs/2605.21193).
## References
1. [Robust dimension free isoperimetry in Gaussian space, arXiv:1202.4124](https://arxiv.org/abs/1202.4124)
2. [The Brunn-Minkowski inequality in Gauss space, CiNii Research record, Inventiones Mathematicae 30(2), 1975](https://cir.nii.ac.jp/crid/1360011146621655040)
3. [The quantitative isoperimetric inequality and related topics, Bulletin of Mathematical Sciences, Springer](https://link.springer.com/article/10.1007/s13373-015-0074-x)
4. [Concentration lecture notes, Section 6.4, Georgia Tech (Lifshits)](https://glivshyts6.math.gatech.edu/Conc-lecture-notes.pdf)
5. [Communications on Pure and Applied Mathematics, Wiley](https://onlinelibrary.wiley.com/doi/10.1002/cpa.70020)
6. [D. Bakry, M. Ledoux (1996). Lévy–Gromov's isoperimetric inequality for an infinite dimensional diffusion generator. Inventiones Mathematicae 123, 259–281.](https://www.math.univ-toulouse.fr/~ledoux/LevyGromov.pdf)
7. [Sudakov, V. N. & Tsirel'son, B. S. (1978). Extremal properties of half-spaces for spherically invariant measures. Journal of Soviet Mathematics 9(1), 9–18. Tel Aviv University research portal](https://cris.tau.ac.il/en/publications/extremal-properties-of-half-spaces-for-spherically-invariant-meas/)
8. [M. Lifshits. Lectures on Gaussian Processes, Springer, 2012](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MV/MVspa/Lifshits%20M.%20Lectures%20on%20Gaussian%20processes%20%28Springer%2C%202012%29%28ISBN%209783642249389%29%28O%29%28133s%29_MVspa_.pdf)
9. [Gaussian Isoperimetric Inequality — Statement & Proof, Androma](https://androma.org/theorems/6759)
10. [C. Borell (1975). The Brunn-Minkowski Inequality in Gauss Space, original PDF](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2023/06/Borell-Subspace-original.pdf)
11. [R. Latała. On Some Inequalities for Gaussian Measures](https://duch.mimuw.edu.pl/~rlatala/papers/gaussicm02.pdf)
12. [Problem sheet 2, Ehrhard inequality, École Normale Supérieure de Lyon](https://perso.ens-lyon.fr/aurelien.garivier/www.math.univ-toulouse.fr/_agarivie/concentration/AGG-exo-2.pdf)
13. [Report 08/2010, Kawohl, on Ehrhard symmetrization proofs](https://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub97.pdf)
14. [S. G. Bobkov (1997). An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Annals of Probability 25(1), 206–214.](https://www-users.cse.umn.edu/~bobkov/papers/1997_AOP.pdf)
15. [High-Dimensional Probability, course page, R. Vershynin, UC Irvine](https://webapps.math.uci.edu/~rvershyn/teaching/hdp/hdp.html)
16. [Concentration and comparison lecture notes, M. Krishnapur, IISc](https://math.iisc.ac.in/~manju/GP/6-Concentration%20and%20comparison%20again.pdf)
17. [The Anisotropic Gaussian Isoperimetric Inequality and Ehrhard Symmetrization, arXiv:2309.03853](https://arxiv.org/abs/2309.03853)
18. [Sharp Gaussian Isoperimetry along a Ricci Flow, arXiv:2605.21193](https://arxiv.org/abs/2605.21193)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Stochastic processes › Continuous-time and continuous-state processes › Gaussian and Wiener processes › Regularity and sample-path properties of Gaussian processes*
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