Determine whether a point lies in a convex hull of points in O(logn)
This is a classic problem in computational geometry, called Polygon Inclusion Problem. Further, you are considering its special case --- Convex Inclusion.

In chapter 4 of this thesis by Michael lail Shamos 1978, you will find that:

Generally,


  Theorem 4.2 (Page 92): Whether a point is interior to a simple $n$-gon can be determined in $O(n)$ time, without preprocessing. 


and for convex polygon,


  Theorem 4.3 (Page 95): The Inclusion question for a convex $n$-gon can be answered in
  $O(\log n)$ time and $O(n)$ space, given $O(n)$ preprocessing time.


The proofs of these two theorems contain the algorithms you are looking for.