Let $\mu,\nu$ be probability measures on $\mathbb{R}^d$ with finite $p$-th moment ($p\in [1,\infty)$) and define the set of couplings by $\mathcal{C}(\mu,\nu)$ i.e. the set of probability measures on $\mathbb{R}^d\times \mathbb{R}^d$ so that the projection onto each component is $\mu,\nu$.
Define an integral functional $I_{p}(\pi,\omega):\mathcal{C}(\mu,\nu)\times \mathbb{S}^{d-1}\to \mathbb{R}$ by
$\displaystyle \begin{equation} I_p(\omega, \pi) =\left(\int_{\mathbb{R}^n\times \mathbb{R}^n}|\langle \omega ,x-y\rangle |^p\pi(dx\times dy)\right)^{1/p} \end{equation}\tag*{}$
where $\langle x,y \rangle$ denotes the standard Euclidean inner product on $\mathbb{R}^n$.
I am interested in proving or disproving that:
$\displaystyle \inf_{\pi\in \mathcal{C}(\mu,\nu)}\int_{\mathbb{S}^{d-1}}I_p(\omega,\pi)^pd\sigma_{d-1}=\int_{\mathbb{S}^{d-1}}\inf_{\pi\in \mathcal{C}(\mu,\nu)}I_p(\omega,\pi)^pd\sigma_{d-1}\tag*{}$
Here, $\sigma_{n-1}$ is the standard volume measure on $\mathbb{S}^{d-1}$.
Background. This is related to sliced Wasserstein distance. The RHS is almost the definition of the sliced W distance (aside from $1/p$ power).
What I tried:
Following this MSE post, I tried to construct a sequence $\pi_n$ such that $I_p(\pi_n, \cdot) \to \textrm{inf}_{\pi}I_p(\pi,\cdot)$ uniformly in $\omega$. However, I was unable to do so. On the way to prove it, I have observed the following facts.
(1) $I_p$ is continuous assuming we give $\mathcal{C}(\mu,\nu)$ the topology of weak convergence. (The proof is a bit long, so I am omitting it here. The Cauchy-Schwartz $|\langle \omega, x-y\rangle | \leq \|x-y\|$ is heavily in use.)
(2) Since $\mathcal{C}(\mu,\nu)$ and $\mathbb{S}^{d-1}$ are compact, $I_p$ is uniformly continuous, and the family $\{I_p(\cdot,\omega)|\omega \in \mathbb{S}^{d-1}\}$ is uniformly equicontinuous.
(3) I wanted to use the above facts to find $\pi_n$, but this won't work. For example, $f(x,y)=|x-y|$ on $[0,1]\times [0,1]$ satisfies the same properties as above, but $\inf_{x}\int_{0}^{1}|x-y|dy\neq \int_{0}^{1}\inf_{x}|x-y|dy$.
(4) If we can find the uniform minimizer $\pi^*$ (i.e. for any $\omega$ and $\pi$, $I_p(\pi^*, \omega)\leq I_p(\pi,\omega)$), then its trivial since we can take $\pi_n=\pi^*$, but I think this is too much to ask for.