Under the assumption, that $n$ points in random order are given, the best algorithm seems to be to construct the generalization of the Delaunay Triangulation to $d$-dimensional Euclidean space; that yields a collection of empty hyper-balls that are defined via $d+1$ of the points; the number of those hyper-balls is $O(n^{\lceil d/2 \rceil})$.  
The bound on the number of empty hyper-balls proves that the convex hull of $n$ points can't have an exponential number of faces like e.g. $O(2^n)$.

From that collection of hyperballs the ones, whose center is outside the convex hull of their defining $k>=d+1$ points, are not inside the convex hull of the $n$ points and are not considered further.

Then one has to check, whether the radius of any of the remaining hyper-balls is at least $1$.

From the efficient construction of the Delaunay Triangulation in higher dimensions, it follows, that the answer to the question is yes.