In this article, we establish some new combinatorial properties of cone types in Coxeter groups. Firstly, we show that for any element $x$ in a Coxeter group $W$ and root $β$ in its inversion set $Φ(x)$, the set of elements $y \in W$ satisfying $Φ(x) \cap Φ(y) = \{ β\}$ is convex in the weak order and admits a unique minimal representative. This is strongly connected to determining the cone type of elements of $W$ and leads to efficient computational methods to determine whether arbitrary elements of $W$ have the same cone type.