Maximum Polyhedron Volume in Given $n$ Points
This was shown to be hard (more precisely $\mathsf{NP}$-hard to approximate to better than exponential in $k$) by Marco Di Summa, Friedrich Eisenbrand, Yuri Faenza, Carsten Moldenhauer, "On largest volume simplices and sub-determinants", arXiv:1406.3512 and SODA 2015. (They talk about the largest volume simplex contained in the convex hull, rather than the one that has the given points as vertices, but it amounts to the same thing since there is always an optimal contained simplex that has the given points as vertices.) They cite earlier hardness results on the same problem by Packer (ref.37 of the arXiv preprint) and Papadimitriou (ref.38).