Testing whether a set of n points in the plane form a convex n polygon in o(nlogn) time
Thats seems unlikely, at least in the comparison/algebraic tree models. Definition first:

A point set $P$ is in convex position if no point of $P$ can be written as a convex combination of the remaining points of $P$.

Now, deciding if a set of $n$ numbers are all distinct takes $\Omega(n \log n)$ time (this is known as UNIQUENESS). Given such a set of $n$ numbers $X$, map them to the set of points
$$
P = \{ (x, x^2) | x \in X\}.
$$
If there is no repeated number, then the points are in convex position.

If there is a repeated number, then this repeated number corresponds to a point that can be written as a convex combination of the remaining points. Namely, the points are not in convex position.

Namely, deciding if a point set is in convex position is as hard as UNIQUENESS.