Given any $d$-dimensional shape $X$, let $V(X)$ be its $d$-dimensional volume, and let $\ell(X)$ be the length of the longest line segment connecting two points of $X$.


Let $\mathcal{S}\_C$ be the set of all $d$-dimensional shapes such that their minimum bounding box is a $d$-dimensional cube $C$. I am interested in quantifying the trade-off between $\frac{V(X)}{V(C)}$ and $\frac{\ell(X)}{\ell(C)}$ over $X\in\mathcal{S}\_C$ (informally, how much $\frac{V(X)}{V(C)}$ can be large while $\frac{\ell(X)}{\ell(C)}$ is small).


**Question:** Can we prove that for $d\gg 1$ and for all $X\in\mathcal{S}\_C$ there exists a constant $c$ such that the following inequality always holds?
$$\left(\frac{V(X)}{V(C)}\right)^{\tfrac1d}\le c\cdot\frac{\ell(X)}{\ell(C)}$$



Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box
This is a bit too long for the comment box, so I'm posting it as an answer.


The worst case scenario is when $X$ is the intersection of a ball of radius $r\ge 1$ with the cube $C=[-1,1]^d$. Indeed, if we take the difference body $\frac{X-X}{2}$ of any body $X$ contained in the cube and of diameter $\ell=2r$, we'll get a body contained in the cube and also in the ball of radius $r$ and the volume will not decrease by Brunn-Minkowski. Also, since any such body contains the unit ball, the standard cube is, indeed, the minimal box for it. Since $\frac{\sqrt n}r X\supset C$, we see that for that body the reverse inequality always holds.


It would be nice to find a decent approximation for the volume of that intersection to see what happens in the regime when $r/\sqrt d$ stays fixed and $d\to\infty$, say.


