Does there exist, for any natural $n$, a probability distribution in $\mathbb{R}^n$ whose projection on any line is a uniform distribution?



A probability distribution in n dimensional space which its projection on any line is a uniform distribution?
No if $n\geq 4$. By translating we may assume that the mean is $0$, and by scaling we may assume that the variance-covariance matrix is the scalar matrix $(1/3)I$. Then all projections have mean $0$ and variance $1/3$, hence are uniform in the interval $[-1,1]$.


So if we take a random projection, the value will also be uniform in the interval $[-1,1]$.


Let $\mathbf x$ be a random variable along such a distribution. Clearly $|\mathbf x|\leq 1$ with probability $1$. The probability that the absolute value of the projection of $\mathbf x$ in a random direction is at least $1-\epsilon$ is an increasing function of $|\mathbf x|$, so it is at most the probability that the projection of a unit vector in a random direction has length at least $1-\epsilon$. This is the same as the proportional of the unit $n-1$-sphere that is in a spherical cap of depth $\epsilon$, which is $\approx \epsilon^{(n-1)/2}$.


On the other hand, because the distribution is uniform, the probability that the absolute value of the projection of $\mathbf x$ in a random direction is at least $1-\epsilon$ must be $\epsilon$. So we have $\epsilon \leq \epsilon^{(n-1)/2}$, and thus $n\leq 3$.


