I'm trying to figure out the following problem:

Let $x_1,\ldots,x_k\in\mathbb{R}^n$ be some points for some $k<n$. Let $\mbox{conv} (x_1,\ldots,x_k)$ be their convex hull. I'm looking for a tight (possibly with an example) upper bound for the number of orthants that such convex hull can intersect with (depending on $n$ and $k$).

I've figured that as the convex hull of two points may intersect with up to $n$ orthants, by induction I can bound the number of intersections of $k$ points with different orthants to be $O(n^{k-1})$. I want to prove that this bound is tight or find a tighter one, hopefully with a concrete example.

Any ideas? Thanks!