balls in arrangements of hyperplanes

2011-04-04T03:25:29Z

The following theorem is from Aronov, Naiman, Pach and Sharir's An invariant property of balls in arrangements of hyperplanes. I would like to state them and then ask if any related problem/theorem is known since the paper was released in 1993.

Let $H$ be a collection of $n$ hyperplanes in $d$-space in general position. For each tuple of $d+1$ hyperplanes of $H$, consider the open ball inscribed in the simplex that they form. Let $B_{k}$ denote the number of such balls intersected by exactly $k$ hyperplanes, for $k=0,1, \cdots, n-d-1$. Then $$B_{k}=\binom{n-k-1}{d}$$

In the paper, they used a very clever combinatorial proof to show such an invariant property.

Here I am trying to find any connected problem to this. For example, is the case when $d=2$ or $3$ a known theorem?

Also I have thought of the cell decomposition property in incidence geometry, which states

Let $L$ be a finite collection of lines in $R^2$, and let $r \ge 1$. Then it is possible to find a set of $O(r)$ lines in the plane, which subdivide $R^2$into $O(r^2)$ convex regions (or cells), such that the interior of each such cell is incident to at most $O(|L|/r)$ lines.

Is this related?

balls in arrangements of hyperplanes - MathOverflow

balls in arrangements of hyperplanes