# The Isoperimetric Inequality

Simon Brendle
Michael Eichmair

Communicated by Notices Associate Editor Chikako Mese

## 1. The Isoperimetric Inequality and the Sobolev Inequality

The isoperimetric problem is one of the oldest and most famous problems in geometry. Its origins date back to the legend of Queen Dido founding the City of Carthage, as told in Virgil’s Aeneid.

In two dimensions, the isoperimetric inequality asserts that a disk has the smallest boundary length among all domains in the plane with a given area.

Here, denotes the area of and denotes the length of the boundary . Note that disks achieve equality in the isoperimetric inequality. Indeed, if is a closed disk of radius in the plane, then and .

Theorem 1 is a special case of a more general inequality which holds in arbitrary dimension.

Here, denotes the volume of and denotes the -dimensional measure of the boundary . Moreover, denotes the open unit ball in and denotes its volume.

The isoperimetric inequality is sharp on balls. To see this, recall that the volume and boundary area of the unit ball in are related by . Hence, if is a closed ball of radius , then and .

Another important inequality related to the isoperimetric inequality is the sharp version of the Sobolev inequality.

The Sobolev inequality plays a fundamental role in the modern theory of partial differential equations. For a function defined on a ball, we have the following variant of the Sobolev inequality.

Note that Theorem 4 implies the Sobolev inequality on (Theorem 3). To see this, we assume that is a smooth function on with compact support. After a suitable rescaling, we may assume that the support of is contained in the open unit ball . We then apply Theorem 4 to the function and send .

Moreover, Theorem 3 implies the isoperimetric inequality (Theorem 2). To see this, we assume that is a compact domain in with smooth boundary. We then approximate the indicator function of by a sequence of nonnegative smooth functions with compact support. To explain this, we fix a smooth cutoff function such that for , for , and for . For each positive integer , we define . If is sufficiently large, then is a nonnegative smooth function on . Moreover,

while

as . Theorem 3 then implies .

In Sections 2 and 3 we present several different proofs of Theorem 4. In Section 2, we sketch how Theorem 4 can be proven using measure transportation. This strategy is due to Gromov and can be implemented in two ways. Gromov’s original approach uses the Knothe rearrangement. An alternative approach, due to McCann and Trudinger, is based on the Monge-Ampère equation. In Section 3, we discuss a proof of Theorem 4 due to Cabré that uses linear partial differential equations and the Alexandrov-Bakelman-Pucci method.

## 2. Proof of Theorem 4 Using Measure Transportation

In this section, we present the measure transportation approach to Theorem 4. By scaling, one can reduce to the special case where . The first step of the proof involves constructing a smooth map from the open unit ball into itself with the following properties:

(i)

For each point , the eigenvalues of the differential are nonnegative real numbers.

(ii)

For each point , the determinant equals .

Suppose that is a map with these properties. We may view as a vector field defined on . Since the eigenvalues of the differential are nonnegative real numbers, their geometric mean can be estimated from above by their arithmetic mean. This gives

at each point in . Since takes values in the unit ball, we know that at each point in . Consequently,

at each point in . In the next step, we integrate over the ball , where . Using the divergence theorem, we conclude that

for each . On the other hand, using again the fact that maps into the unit ball, we obtain for each and each point . This implies

for each . Sending , one obtains

Using the normalization , it follows that

as desired.

It remains to construct a map that satisfies the conditions (i) and (ii) above. Gromov’s proof in 19 is based on the Knothe rearrangement 16. This construction gives a smooth map from the open unit ball to itself with the following properties:

For each point , the differential is a triangular matrix and the diagonal entries of are nonnegative.

For each point , the determinant equals .

Clearly, the Knothe map satisfies the conditions (i) and (ii) above.

Let us sketch the construction of the Knothe map. For simplicity, we consider the special case . The Knothe map has the form for . The function maps the interval to itself and satisfies

for each . For each , the function maps the interval to the interval and satisfies

for each .

We next describe an alternative approach, due to McCann and Trudinger, which is based on a different choice of the map . The key step in this approach is to solve a suitable boundary value problem for the Monge-Ampère equation. It was shown by Caffarelli 8 and Urbas 20 that there exists a convex function with the following properties:

The function is smooth and solves the Monge-Ampère equation

at each point in .

maps to itself.

We now define to be the gradient map of , so that for each . At each point , the differential is a symmetric matrix with nonnegative eigenvalues, and the determinant equals . Therefore, the gradient map satisfies the conditions (i) and (ii) above.

## 3. Proof of Theorem 4 Using the Alexandrov-Bakelman-Pucci Method

In this section, we describe a proof of Theorem 4 using the Alexandrov-Bakelman-Pucci technique. This technique plays a central role in the theory of partial differential equations, where it is used to prove a-priori estimates for elliptic partial differential equations in nondivergence form. Cabré 7 showed that the Alexandrov-Bakelman-Pucci technique can be used to give an alternative proof of the isoperimetric inequality. His argument can be adapted to give a proof of the Sobolev inequality.

By scaling, one can reduce to the special case where . This normalization ensures that one can find a function with the following properties:

The function is twice continuously differentiable and solves the linear partial differential equation

at each point in .

The function satisfies the Neumann boundary condition

at each point .

The existence and regularity of follow from the standard theory of linear elliptic partial differential equations of second order.

Let denote the gradient map of , so that for each . Let denote the set of all points with the property that and the Hessian is weakly positive definite.

Clearly, at each point in . The partial differential equation for implies that

at each point in . Applying the arithmetic-geometric mean inequality to the eigenvalues of the Hessian of , one obtains

at each point in . Using the change-of-variables formula, one can estimate the measure of the image . This gives

On the other hand, it can be shown that the set contains the open unit ball . To see this, suppose that a point is given. It follows from the Neumann boundary condition for that the function attains its minimum at an interior point . The first- and second-order conditions at the minimum point imply that and the Hessian is weakly positive definite. Thus, and .

Since contains the open unit ball , we obtain

Combining 1 and 2 gives

In view of the normalization, it follows that

This completes the proof of Theorem 4.

## 4. The Sobolev Inequality and the Isoperimetric Inequality on a Hypersurface in $\mathbb{R}^{n+1}$

We next discuss how the Sobolev inequality and the isoperimetric inequality can be generalized to hypersurfaces in . It is particularly natural to study this question for minimal hypersurfaces.

To explain the notion of a minimal hypersurface, we first recall the definition of the mean curvature. Suppose that is a compact smooth hypersurface in (possibly with boundary), and let be a point on . We may locally write as a level set , where is a smooth function which is defined on an open neighborhood of and satisfies . The unit normal vector field to is given by . Moreover, the mean curvature of is given by

It turns out that this definition depends only on the hypersurface and the choice of orientation. It does not, however, depend on the choice of the defining function .

The notion of mean curvature is closely related to the formula for the first variation of area. To explain this, suppose that is a smooth vector field on . If has non-empty boundary, we assume that the vector field vanishes along the boundary of . We consider the deformed hypersurfaces , where is a small real number and the maps are defined by for . In other words, we deform the hypersurface with a velocity given by the vector field . Since vanishes along the boundary of , this deformation leaves the boundary of unchanged. With this understood, the first order change in the area is given by

where denotes the mean curvature of (see, e.g., 12, Chapter 1, Section 1).

In particular, if is a minimal hypersurface, then is a critical point of the area functional.

There are many examples of minimal surfaces in . The most basic ones are the plane, the catenoid

and the helicoid

In 1921, Carleman 9 showed that the isoperimetric inequality holds for two-dimensional minimal surfaces in that are diffeomorphic to a disk.

Note that this inequality is sharp. Carleman’s proof of Theorem 7 uses techniques from complex analysis.

Theorem 7 raises the question whether the isoperimetric inequality holds for minimal surfaces of arbitrary dimension and topology. In the 1970s, Allard 1 and Michael and Simon 18 proved a general Sobolev inequality which holds for arbitrary hypersurfaces in Euclidean space (and, more generally, for submanifolds of arbitrary codimension). Their arguments are based on the monotonicity formula in minimal surface theory together with covering arguments. More recently, Castillon 10 gave an alternative proof based on techniques from optimal transport. However, these works do not give a sharp constant. In 2019, the first-named author proved a sharp version of the Michael-Simon-Sobolev inequality.

In particular, if is a minimal hypersurface, then the mean curvature term vanishes and we can draw the following conclusion.