I accidentally came across a curious inequality for functions of two variables. I would like to know if this inequality is a special case of a more general result but I was unable to find a reference. It would also be interesting if applications can be found for this inequality.
Theorem 1 For any nontrivial function on such that and is compact, we have
Here we define the integrand to be zero if .
In particular, if is compactly supported, then
Proof: By Sard’s theorem, for almost all , is a smooth level curve which must be closed (but may be disconnected). Fix such for the moment. It is easy to compute that the curvature of the level curve is given by
where is the unit tangent and .
Expanding, we have
On the other hand, by the coarea formula we have
where we have used Fenchel’s inequality ([dC] p.199) in line 4.
Remark 1 We record some simple observations here.
By the equality case of the Fenchel’s inequality, it can be seen that the equality holds if and only if for almost every , the level curve is a (connected) closed convex curve or . Clearly in this case, by choosing a suitable sign, we can remove the absolute value in the integral.
1. Higher dimension and a more intrinsic version
We now investigate the higher dimensional analogue. By the result of this paper (Theorem 1) of Chen, it can be similarly shown that an inequality for a certain integral involving the Hessian of when restricted to the tangent spaces of the level sets holds. However, the expression is not as explicit. Indeed, it is not hard to see that under the same assumptions,
where and is the area of the unit -sphere. Here is defined as follows: choose an orthonormal basis for ( on for almost all ), then we define where . This is easily seen to be well-defined.
Question: I wonder if notions like this has been studied by others, and whether has a more computable form.
Now we prove a more intrinsic version of the inequality which does not involve . Let , then by AM-GM inequality,
where is the unit normal to the level surfaces.
So (2) can be expressed more intrinsically as the following:
Theorem 2 For any nontrivial function on such that and is compact, we have
Here we define the integrand to be zero if and recall that is the area of the unit -sphere in and .
In particular, if is compactly supported,
Let us also record a simple observation. The equality holds if and only if each level set is convex and connected, and is constant along the non-degenerate level surfaces and is umbilical, i.e. for some function and for tangential to the level surfaces. But these two conditions imply that the level surfaces are all umbilical, and they are either planes or hyperspheres. But since all level surfaces are complete and compact by our assumptions, so they must be hyperspheres.
In particular, if is a monotone radial function, then the equality holds. Indeed, by direct computation, and where is the sphere of radius in and is the Euclidean metric. By direct computation, the integrand becomes . Since the Euclidean volume form is given by and has a definite sign, it is easy to see that the equality holds.