# Volume of bounded regions in hyperplane arrangements

I am given a hyperplane arrangement $\mathcal{H}_0$ in $\mathbb{R}^n$ and a function $\phi \colon \mathbb{R}^n \to \mathbb{Q}.$ I choose any enumeration on the set of primitive vectors (i.e. vectors whose entries are integers and have gcd = 1) $v_1, v_2, \cdots$. Then, I recursively form the hyperplane arrangments $\mathcal{H}_{i+1} = \mathcal{H}_i \cup \{l_{i+1}\}$ where $l_{i+1}$ is the line defined by $\langle v_{i+1}, x\rangle = -\phi(v_{i+1})$

I am interested in how the sum of the volumes of the bounded regions increase as I let this process go to infinity (i.e. $i \to \infty$). I would like something like a convergence criterion in terms of the function $\phi$.

Does anyone have any ideas? Or has a reference in mind which can help?

• what kind of functions $\phi$ do you have in mind? Can you say something for $n=2$ that would help us visualize? – Graham Denham Jan 26 '15 at 0:18
• If n= 2, consider for example $\phi(x,y) = \begin{cases} \frac{-xy}{x+y} \text{ if }x,y \geq 0 \\ min \{0,x,y \} \text{ otherwise } \end{cases}$. This is a concave functions, hence has bounded stability set, hence the regions converge to a convex set, which is exactly the stability set of this function. – cata Jan 26 '15 at 13:29