Given a set of $n$ points in $\mathbb{R}^d$, is there an algorithm to determine if the convex hull contains the unit ball centered at the origin in polynomial time (in both $n$ and $d$)? The convex hull itself might have an exponential number of facets so we cannot afford explicitly to compute it.
My main interest is not in computer precision so we can make whatever assumptions help avoid that in relation to the points themselves (for example they only have integer coordinates).
The problem is NP hard. Here is a proof sketch.
The problem is to determine if there is a point $y$ with $\|y\|=1$ outside of the convex hull of given points $x_1,\dots, x_n$. Note that such point exists if and only if there is hyperplane at distance less than $1$ from the origin such that all points $x_1, \dots, x_n$ and $0$ lie on one side of the hyperplane (consider a hyperplane $\pi$ separating $x_1,\dots, x_n, 0$ and $y$).
So the problem can be stated as follows: is there $a\in {\mathbb R}^d$ s.t.
Note that this problem is equivalent to the following well-known problem:
We are given a convex polytope $\cal P$, a positive definite matrix $A$ and a number $t$, find $x\in \cal P$ such that $x^T A x > t$. ($\cal P$ is described by a system of linear equations).
The optimization version of this problem is:
Quadratic Programming (Non-convex Linearly Constrained Quadratic Programming with Positive Definite Matrix). We are given a convex polytope $\cal P$ and and a positive definite matrix $A$, find $x\in \cal P$ that maximizes $x^T A x$.
This problem is known to be NP hard.
It is even NP-hard to optimize $x^T A x$ when $\cal P$ is the unit cube $\{(b_1, \dots, b_d): -1\leq b_i \leq 1\}$ (this problem is known as Integer Quadratic Programming with Positive Definite Matrix). In particular, the MAX CUT problem is a special case of this problem. Let $G$ be a graph on $n$ vertices and $L$ be its Laplacian. Then $\max_{x\in\{\pm 1\}^n} x^T L x$ is equal to the size of the maximum cut in $G$ ($L$ is positive semi-definite, not positive definite, but this difference is not important; e.g. we can consider $L'=L+\varepsilon I$ with very small $\varepsilon$). The NP-hardness of MAX CUT was proved by Karp in
Richard M. Karp (1972). Reducibility Among Combinatorial Problems. In R. E. Miller and J. W. Thatcher (editors). Complexity of Computer Computations. New York: Plenum. pp. 85–103.
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.
The situation doesn't change, if only the points on the convex hull shall be taken into account; this is so, because the points on the convex hull can also be efficiently determined via a generalized gift-wrapping algorithm and then the generalized Delaunay Triangulation can be constructed for those points and finally the largest empty hyper-ball can be determined as described before.
If the unit ball is centered at the origin, then one has to check, whether distance of the the faces of the convex hull that were reported by the gift-wrapping algorithm, is not less than $1$.
