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We show that the problem of finding the simplex of largest volume in theconvex hull of $n$ points in $\mathbb{Q}^d$ can be approximated with a factorof $O(\log d)^{d/2}$ in polynomial time. This improves upon the previously bestknown approximation guarantee of $d^{(d-1)/2}$ by Khachiyan. On the other hand,we show that there exists a constant $c>1$ such that this problem cannot beapproximated with a factor of $c^d$, unless $P=NP$. % This improves over the$1.09$ inapproximability that was previously known. Our hardness result holdseven if $n = O(d)$, in which case there exists a $\bar c\,^{d}$-approximationalgorithm that relies on recent sampling techniques, where $\bar c$ is again aconstant. We show that similar results hold for the problem of finding thelargest absolute value of a subdeterminant of a $d\times n$ matrix.

On largest volume simplices and sub-determinants