For a \(d\)-simplex \(S\subset E^d\), denote by \(v_1,\dots, v_d\) the vectors determining the edges of \(S\) starting at its vertex \(0\), say. It is clear that the \(d\)-parallelotope \(P\) with vertex \(0\) and edges at \(0\) determined by the vectors \({1\over d}v_1,\dots, {1\over d}v_d\) is a subset of \(S\) and has \(d!/d^d\) times the volume of \(S\). The author shows that no \(d\)-parallelotope contained in \(S\) can have larger volume. He describes all \(d\)-parallelotopes of largest volume contained in \(S\) and shows that all of them are inscribed in \(S\). A final discussion presents motivations for such questions as well as related results and open problems.