On Extremal Volume Projections of the Simplex and the Cube

Let $\Delta_n$ and $Q_n$ denote the regular $n$-simplex of side length $\sqrt{2}$ embedded in $\mathbb{R}^{n+1}$ and the volume one cube in $\mathbb{R}^n$, respectively. We derive a closed-form formula for the hyperplane volume projections of $\Delta_n$, which also yields the directions achieving the extremal volume. Moreover, we revisit the problem of extremal planar projections of $Q_n$. In addition, we present generalizations within the framework of $L_p$-projection bodies.