Let $C\subset\mathbb{R}^n$ be a cone with vertex at the origin, aperture $\theta$ and height $h$. Since a cone is a convex region in $\mathbb{R}^n$ we know there is a parallelotope $P$, completely contained in $C$ such that

$Vol_n(P)\geq n^{-n}Vol_n(C),$

where $Vol_n(A)$ is the $n$-dimensional volume of $A$. This result is Lemma 8 of A compactness theorem for Affine equivalence-classes of convex regions by Macbeath. (note the statement of the Lemma has a typo - the inequality is stated the wrong way round!)

I was wondering what happens if we fix a vertex of the parallelotope to the origin. That is if $\mathcal{P}$ is the family of parallelotopes completely contained in $C$ with a vertex at the origin.

What are lower bounds on

$\sup_{P\in\mathcal{P}} Vol_n(P)/Vol_n(C)?$