 Maximal cylinder and box in an elliptic paraboloid

For the rotational elliptic paraboloid $z=H(1-(x^2+y^2)/R^2)$ above $z=0$, allow inscribed shapes to touch the boundary. A coaxial cylinder of radius $r$ and height $h$ obeys $h+H{r}^2/R^2=H$. Applying a Lagrange multiplier to $\pi r^{2}h$ gives $h=H{r}^2/R^2$, hence $h=H/2$ and $r=R/\sqrt{2}$.

A centred rectangular parallelepiped with horizontal side lengths $a,b$ and height $c$ obeys $c+H(a^2+b^2)/(4R^2)=H$. The Lagrange multiplier equations for $abc$ give $a=b$ and $c=H{a}^2/(2R^2)$, hence $a=b=R,c=H/2$. These yield the displayed maximal volumes; both positive stationary shapes are interior optima among the dimension variables, while their degenerate boundary shapes have zero volume. The cylinder's volume is $\pi /2$ times the box's. If strict containment in the open region is required, these are suprema approached by slightly smaller shapes rather than attained maxima.