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.
Also, if
is
and all its critical points are non-degenerate (i.e. Morse function), then by Morse’s theory, if there is a critical point in
with index
, then the inequality (1) is strict.
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.