Given a continuous random variable $X$ with the cdf $F_X(x)$, I want to know whether there exists a random variable $\mathbf{Z}$ uniformly distributed in a geometry region $\mathscr{Z}_n$ in $\mathbb{R}^n$ such that any one-dimensional marginal distribution $F_{\mathscr{Z}_n}^1$ of $\mathbf{Z}$ is close to the distribution of $X$. In addition, an asymptotic perspective is also welcome. For example, it is also meaningful to know whether $F_{\mathscr{Z}_n}^1=F_X$ is possible if $n$ tends to $+\infty$.

Some useful facts are given as follows:

1. A normal distribution can be approximated by a uniform random variable distributed in a hypersphere.
2. A negative exponential distribution can be approximated by a uniform random variable distributed in a simplex.

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