Frank Morgan

In geometry the most fundamental inequalities are isoperimetric inequalities. In this post we will focus on dimension two, although all of the results extend to higher dimensions. In R², the perimeter and area of a region satisfy

(1)            P ≥ (4πA)^1/2,

with equality for a round disc. On the unit sphere, for 0 < A < 4π,

(2)            P ≥ (A(4π-A))^1/2,

with equality for a geodesic disc.

In analysis, the most fundamental inequalities relate functions and often their derivatives. For example, the Cauchy-Schwarz Inequality says that for two smooth functions f and g,

∫ f g  ≤  (∫f²)^1/2 (∫g²)^1/2,

with equality when one of the functions is a multiple of the other. The L¹ Sobolev inequality says that for a smooth function of compact support on R²,

(3)            ∫  |grad f| ≥ c (∫f²)^1/2.

To bring to bear the tools of functional analysis on geometry, one would like to find functional versions of isoperimetric inequalities. One such inequality equivalent to (1) is that for any integrable function f from R² to {0,1},

(4)            ∫ |grad f|  ≥ (4π ∫f )^1/2.

If f is the characteristic function of a region (of finite area), the integral on the right gives the area, and the integral on the left, with grad f suitably interpreted where f is not smooth, gives the perimeter. One would much prefer an inequality for all smooth functions f. Unfortunately, (4) does not hold for all smooth functions or even for piecewise constant functions such as half the characteristic function of the unit disc (or smooth approximations thereof). It is necessary to add some sort of penalty for fractional values of f. Federer and Fleming [FF, Thm. 6.4] and Maz’ya [M] obtained the right generalization, a sharp form of the L¹ Sobolev inequality (3): for all smooth functions of compact support from R² to R,

(5)            ∫ |grad f|  ≥   (4π∫f²)^1/2,

with equality approached as f approximates any multiple of the characteristic function of a round disc.

In the Gauss plane, defined as R² with Gaussian density (1/2π)exp(-x²/2), the least-perimeter way to enclose given area A is a half-plane with perimeter P = I(A), i.e., for any region, I(A) ≤ P. The simplistic functional analog for characteristic functions of regions is

(6)            I(∫f) ≤ ∫ |grad f| ,

but again this fails for smooth functions and even for piecewise constant functions. Indeed, since the Gauss plane has finite total weighted area 1, any area 0 < A < 1 may be achieved by taking the whole plane with some fractional density, and the right-hand side of (6) vanishes. Again it is necessary to add some sort of penalty for fractional values of f. The right generalization was provided by Bobkov [Bo]: for any smooth function f from R² to [0,1],

(7)            I(∫f) ≤ ∫ (I(f)² + |grad f|²)^1/2) .

Since I(0) = I(1) = 0, (7) reduces to (6) for characteristic functions. A version of this inequality, the so-called Gaussian logarithmic Sobolev inequality, played an essential role in Perelman’s proof of the Poincaré conjecture.

Barthe [Ba, Thm. 5]  provided an analog for the unit sphere with constant density, unit area, and least-perimeter function I(A):

(8)            I(∫f) ≤ (∫ I(f)²)^1/2 + ∫|grad f| ,

which again is sharp for characteristic functions of isoperimetric regions, although in this case it is not sharp for constant functions.

Acknowledgements. I would like to thank Emanuel Milman, Mario Milman, and Christian Houdre, organizers of the 2009 international workshop on “Concentration, functional inequalities, and isoperimetry,” and acknowledge partial support from a National Science Foundation research grant.

References

[Ba]  F. Barthe, Log-concave and spherical models in isoperimetry, Geom. Funct. Anal. 12 (2002), 32-55.

[Bo]  S. G. Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space, Ann. Probab. 25 (1997), 206-214.

[FF]  H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. 72 (1960), 458-520.

[M]  V. G. Maz’ya, Classes of domains and embedding theorems for function spaces, Sov. Math. Dokl. 1 (1960), 882-885. Translation from Doklady Akad. Nauk SSSR 133 (1960), 527-530.

Note added 30 May 2013. Gromov ([G, p. 126]; see [Mo, §10.5] and [CdL, (5.10)]) provides and traces to Brunn’s 1887 inaugural dissertation [B] the Sobolev inequality (5) and its generalization to any norm on the left-hand side, with equality approached as f approximates any multiple of the characteristic function of a Wulff shape.

[B]  H. Brunn, Über Ovale und Eiflächen, Inaugural dissertation, München, 1887.

[CdL]  Daniel Cibotaru and Jorge de Lira, A note on the area and coarea formulas, arxiv.org/abs/1305.2968.

[G]  M. Gromov, Isoperimetric inequalities in Riemannian manifolds; Vitali D. Milman and Gideon Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lec. Notes in Math. 1200, Appendix I, 114-129, Springer-Verlag, 1986.

[Mo]  Frank Morgan, Riemannian Geometry: a Beginner’s Guide, 2nd ed., 1998.

Note added 17 April 2019. It would be nice to have such functional inequalities for the double and multiple bubble problem. Back in 2010 I posted some incorrect ideas, which I have since taken down.