The following iterative algorithm is perhaps fast:
Set $P=d_i$ where $d_i$ is of minimal norm among $d_1,\dots,d_n$.
Iterate the following loop:
Let $j=j(P)$ be an index such that $\alpha=\langle P,d_i-P\rangle/\sqrt{\langle d_i,d_i\rangle}d_i-P,d_i-P\rangle}, i=1,..n$ is minimal for $i=j$.
If $\alpha\geq 0$ then $P$ is at minimal distance. (If $\alpha>-\epsilon$ for small $\epsilon$ you are very close to the minimum.)
Otherwise replace $P$ with the point closest to the origin of the line joining (the old point) $P$ to $d_j$.
End of Loop.
In dimension $d=2$ and if the origin is not in the convex hull, this algorithme will ultimately only involve the two endpoints among $d_1,\dots,d_n$ of the edge realising the minimum (generically).
