Combinatorics of cone types in Coxeter groups

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 $\beta$ in its inversion set $\mathrm{\Phi }(x)$, the set of elements $y\in W$ satisfying $\mathrm{\Phi }(x)\cap \mathrm{\Phi }(y)=\{\beta \}$ 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.