Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$. For a linear subspace $E\subset \mathbb{R}^n$, let $\mu_E$ denote the marginal of $\mu$ on $E$. The usual orthogonal complement of $E$ is denoted $E^{\perp}$.

Suppose $(E_i)_{i=1}^k$ are linear subspaces of $\mathbb{R}^n$ such that

- $\cap_{i=1}^k(E_i \cup E_i^{\perp})=\{0\}$; and
- $\mu$ is equal to the mixture $\mu = \sum_{i=1}^k p_i \mu_{E_i}\otimes \mu_{E_i^{\perp}}$, where $p_i >0$ for each $i = 1, \dots, k$.

Does $\mu$ have finite logarithmic moment: $\int_{\mathbb{R}^n} \log(1+ \|x\|)d\mu(x) <\infty$?

This is a somewhat simplified version of a question I posed in this manuscript. If the answer to the question is affirmative, then it will imply $\mu$ is Gaussian. Some further remarks are given in the appendix of that manuscript. It seems like a nice problem in its own right, so I thought I'd post here.