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The problem of finding a  -simplex of maximum volume in an arbitrary  -dimensional  -polytope, for arbitrary  , was shown by Gritzmann et al. [GKL] in 1995 to be -hard. They conjectured that the corresponding problem for  -polytopes is also -hard. This paper presents a unified way of proving the -hardness of both these problems. The approach also yields -hardness proofs for the problems of finding  -simplices of minimum volume containing  -dimensional  - or  -polytopes. The polytopes that play the key role in the hardness proofs are truncations of simplices. A construction is presented which associates a truncated simplex to a given directed graph, and the hardness results follow from the hardness of detecting whether a directed graph has a partition into directed triangles.

NP -Hardness of Largest Contained and Smallest Containing Simplices for V- and H-Polytopes