Given the vertices of a convex polytope, how can we construct its half-space representation?

The problem you identify is called the facet enumeration problem in the literature: Given the vertices, find a description of the facets. There has been quite a bit of work on this. For $n$ points in $d$ dimensions, $O(n^{\lfloor d/2 \rfloor})$ is achievable, and aymptotically worstcase optimal. But this is a theoretical result. The work of Avis & Fukuda, to which Igor refers, is quite practical, achieving a complexity of $O(d^{O(1)} n M)$ where $M$ is the size of the output description. Here is one reference:

D. Bremner, K. Fukuda, and A. Marzetta. "Primal-dual methods for vertex and facet enumeration." Discrete Comput. Geom., 20(3):333 – 357, 1998. (Citeseer link, which includes a PDF download link.)

If you are interested in software, permit me to also suggest polymake:
          

If you want software, there is Komei Fukuda's cdd et al.