TITLE: Deciding membership in a convex hull
QUESTION [18 upvotes]: Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in the obvious way. I have two questions:

Is there a different (or more efficient algorithm) for this? If not, there ought to be a simple reduction from linear programming to the above problem as well. What is it?

Are there interesting families of instances for which the problem can be solved significantly faster than by means of linear programming, e.g., nearly linear time in $m+n$.

REPLY [4 votes]: "Are there interesting families of instances...nearly linear time..."

For the record it should be mentioned that if the points $v_1,\ldots,v_n \in \mathbb{R}^d$
are chosen randomly, under a variety of distributions and within a
variety of regions (e.g., balls, hypercubes),
then the number of vertices of the hull is $O(n^{1/3})$ in the plane
and $O(\log^{d-1}n)$ in dimension $d$.
A recent reference, which cites many of the original papers is:

Sariel Har-Peled, "On the Expected Complexity of Random Convex Hulls." 2011.
  (arXiv abstract link).