Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube *covers* $P$ if $\operatorname{conv}(Q) \supseteq P$ where $\operatorname{conv}(Q)$ denotes the convex hull of $Q$. How can one compute a minimal set $Q$ that covers $P$? Trivially, a minimal set $Q$ satisfies $|Q| \le |P|$ and $|Q| \le 2^n$. The minimal size could be computed by appealing to decision procedures for the first-order theory of the reals, but is there a smarter way?



convex polyhedron in the unit cube
This is not a direct answer, but a closely related problem is known to be NP-hard:



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> Das, Goodrich.
>  "On the Complexity of Approximating and Illuminating Three-Dimensional Convex Polyhedra."
>  1995. ([ACM link](http://dl.acm.org/citation.cfm?id=645930.672852); [PDF download](http://ranger.uta.edu/~gdas/websitepages/preprints-papers/approx-illum.pdf))
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This paper establishes several results, including this:



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> **Theorem 4.2**. The problem of fitting a polyhedron with a minimum number of faces between 
>  two given nested convex polyhedra is NP-hard. 
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This question was first posed by Victor Klee, and I coauthored a paper that provided an
efficient algorithm in $\mathbb{R}^2$. But the above result shows it is already intractable
in $\mathbb{R}^3$. I do not remember the Das-Goodrich proof well enough to know if it can achieve the
same result with the outer polyhedron a cube.


There are many approximation algorithms available, as this is an important practical problem.
For example:



> 
> Mitchell, Suri. "Separation and approximation of polyhedral surfaces."
>  In *Proc. 3rd ACM-SIAM Sympos. Discrete Algorithms*, pages 296-306, 1992. ([CiteSeer link](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2500))
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