Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance matrix $\Sigma$). For a scalar $b \in \mathbb R$ and nonzero vector $w \in \mathbb R^m$, let $$ R_k(w,b) := \mathfrak R[p_k](w,b) := \int_H p_k(x)\,ds(x)=\int_{\mathbb R^m}p_k(x)\delta(b-x^\top w)\,dx, $$ be the Radon transform of $p_k$ w.r.t the hyperplane $H:=\{x \in \mathbb R^m \mid x^\top w = b\}$, with area element $ds$, and $\delta$ is the Dirac distribution. Let $t \ge 0$ and $\pi_1 \in (0,1)$ and $\pi_2=1-\pi_1$.
Question 1. Is there a simple analytic / algebraic description of the set of pairs $(w,b) \in \mathbb R^m \times \mathbb R$ with $w \ne 0$ such that $$ \pi_1 R_1(w,b+t) = \pi_2R_2(w,b-t)\label{1}\tag{1}\,? $$
Question 2. In the special case where $\pi_1=\pi_2=1/2$, does \eqref{1} have any geometric intepretation, say in terms of optimal transport ?
Now define $\widetilde{R}_k(w,b) := \mathfrak{R}[x \mapsto x_1 p_k(x)](w,b) = \int_{x^\top w \,=\, b}x_1p_k(x)\,ds(x)$, where $x_1$ is the first component of $x$.
Question 3. Same questions as Question 1 and Question 2 but with $R_k$ replaced with $\widetilde{R}_k$.
