Let $h:\mathbb{R}\rightarrow\mathbb{R}$ be Riemann integrable and $\vec{v}\in\mathbb{R}^n$ a constant vector. Show there exists a countinous function $g:\mathbb{R}_+\rightarrow\mathbb{R}$ such that
$$g(||\vec{v}||^2)=\int_B h(\vec{v}\cdot \vec{x}) d\vec{x}$$ where $B$ is the unit ball.
This question was originally in 3-dimensions but seems fair to assume it's true for arbitrary $n$. It's somewhat intuitively true since for two vectors of the same length, the integral should be the same due to symmetry (And I guess it could be formalized using an orthogonal transformation).
I thought about maybe using a change of variables, but the transformation $\vec{x}\rightarrow \vec{v}^T\vec{x}$ isn't $\mathbb{R}^n\rightarrow\mathbb{R}^n$.
Any help is appreciated!