TITLE: Optimizing the gradient norm on the unit sphere
QUESTION [5 upvotes]: Let $ \Bbb S^{d-1}=\{(x_1,\cdots ,x_d): x_1^2+ \cdots +x_d^2=1\}\subset \Bbb R^d$ be the unit
sphere. Let $\nabla u= (\partial_{x_1}u,\cdots, \partial_{x_d}u)$ be the gradient of a function $u\in C_c^\infty(\Omega)$ with $\Omega \subset \Bbb R^d$ open. For $e\in \Bbb S^{d-1}$, we write $$\nabla u(x)\cdot e = \partial_{x_1}u(x) e_1+\cdots+\partial_{x_d}u(x) e_d.$$
Then, for $p\geq 1$, what is the following quantity?
\begin{split}
I(u) =\sup_{e\in \mathbb{S}^{d-1}}\int_\Omega |\nabla u(x)\cdot e|^pdx 
\end{split}
I have tried to use the fact that for all $z\in \Bbb R^d$ we have, $$|z|=\sup_{e\in \mathbb{S}^{d-1}}|z\cdot e|$$ And only got the upper estimate $$I(u)\leq \int_\Omega |\nabla u(x)|^pdx$$
In fact I am expecting an equality here.

REPLY [2 votes]: LONG COMMENT: My collaborators and I have in fact studied the function
$$ x\in \mathbb{R}^d \mapsto \left(\int_{\mathbb{R}^d} |x\cdot\nabla f|^p\,dx\right)^{1/p} $$
quite extensively, for example in this paper. Here are a few highlights:

It defines a norm on $\mathbb{R}^d$. The unit ball of this norm is called the $L^p$ polar projection body of the function $f$. The unit ball of the dual norm is called the $L^p$ projection body. This generalizes the definition of the $L^p$ projection and polar projection bodies of a convex body, which were studied here.
If we denote the norm by $\|\cdot\|_{f,p}$, then the sup of $\|e\|_{f,p}$ over all unit vectors $e$ is half the diameter of the polar projection body, where the diameter is defined to be the maximum distance between two parallel supporting hyperplanes.
Also, note that the quantity
$$ \int_{S^{d-1}} \|e\|_{f,p}^{-n}\,de = \int_{S^d}\left(\int_{\mathbb{R}^d} |e\cdot\nabla f|^p\,dx\right)^{-n/p}\,de $$
is the volume of the polar projection body. Although this does not define a norm on any function space, we proved that it satisfies a sharp affine Sobolev inequality. We call the inequality affine, because this quantity is invariant under the action of $SL(d)$ acting on $\mathbb{R}^d$. It is also not hard to show that this inequality implies the standard sharp Sobolev inequality on $\mathbb{R}^d$ that uses the standard Euclidean norm.

A paper that discuss this function, when $f$ is a probability density function can be found here:
E. Lutwak, D. Yang, G. Zhang. Moment-entropy inequalities, Annals of Probability 32 (2004) 757-774.