I have a $d$-dimensional state $\rho$. Is there any way to find the (possibly not unique) trace distance to the nearest pure state: $$ \min_{|\psi\rangle} \,\,\lVert \rho - |\psi\rangle\langle \psi| \rVert_1 $$
where $\lVert A\rVert_1 = \mathrm{tr}\,\sqrt{A^\dagger A}$. I am not concerned with actually identifying $|\psi\rangle$. An answer that finds the maximum fidelity with a pure state is equally good for my purposes.
For instance, with $d=2$ we can use the Bloch sphere picture to solve this problem: if we represent $\rho$ and $|\psi\rangle \langle \psi|$ as \begin{align} \rho &= \frac{I + \vec{r}\cdot \vec{\sigma}}{2}\\ |\psi\rangle \langle \psi| &= \frac{I + \vec{s}\cdot \vec{\sigma}}{2} \quad, \quad \lVert \vec{s}\rVert=1 \end{align}
where $\vec{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$ is the vector of Pauli operators, then the closest point $\vec{s}$ on the surface of the sphere (pure state) to a point internal to the sphere (mixed state) will lie along the ray formed by $(0,0,0)$ and $\vec{r}$, so the desired pure state is just $$ \vec{s}= \frac{1}{\lVert \vec{r} \rVert} \vec{r} $$
and the corresponding distance is $d(\rho, |\psi\rangle\langle\psi|) = 1 - \lVert \vec{r} \rVert$. However I don't really know how to extend this kind of approach for $d>2$ because I can't really wrap my head around the geometry of higher dimensional state spaces.