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.

 1.  $\langle a, x_i\rangle \leq 1$ (that is, all $x_i$ lie in the half-space $\{x: \langle a, x\rangle\leq 1\}$);
 2. $\|a\| > 1$.

Note 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:

> We are given a convex polytope $\cal P$ and and a positive definite matrix $A$, find $x\in \cal P$ that maximizes $x^T A x$. ($\cal P$ is described by a system of linear equations).

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 $\{(b_1, \dots, b_n): -1\leq b_i \leq 1\}$.