Consider $n-$dimensional Euclidean ball centred at 0 with radius $\sqrt{n}$. We want to show that the uniform distribution $X$ in this ball is sub-gaussian and $||X||_{\psi_2}<C$ where $C$ is absolute constant.

Clarify: $X$ is subgaussian if $\langle X,x \rangle$ is subgaussian for any $x \in \mathbb{R}^n$ and $||X||_{\psi_2}=||\sup_x\langle X,x\rangle||_{\psi_2}$ where sup is over all unit vector $x$.

Attempt: Uniform distribution on ball can be represented by $R,\varphi_1,..,\varphi_{n-1}$ jointly where $R$ is a uniform distribution on $[0, \sqrt{n}]$ representing radius, $\varphi_i$ representing the angles in spherical coordinates and they are uniform on $[0,\pi]$. All these variables are independent.

By symmetry, I only need to show $||\langle X, (1,0,0,0,...)\rangle||_{\psi_2}=||X_1||_{\psi_2}=||R\cos\varphi_1||_{\psi_2}<C$. Then it is not clear to me how to proceed