A sleeping bag for a baby snake in $d$ dimensions (no, really) is a subset of $\mathbb{R}^d$ which can cover (via translation and rotation) every (piecewise-smooth for concreteness) curve of unit length in $\mathbb{R}^d$. For $d=2$ this was posed by Moser, and Stewart asked about the $d=3$ case; the above-linked paper by Hastad/Linusson/Wastlund constructs low-volume sleeping bags in all dimensions $\ge 3$.

A couple of my colleagues and I were talking about the relationship (if any) between the $d=2$ and $d=3$ versions of the problem. Roughly speaking, suppose $X$ is a "reasonably small" $3$-sleeping bag; must some cross section of $X$ be a $2$-sleeping bag? We discussed it for a bit, and it's not obvious to us which of the following (if either) is actually stringent: requiring that some cross section is a sleeping bag, or that no cross section is a sleeping bag.

To be precise:

Let $v$ be the infimum of the volumes of $3$-sleeping bags. Is it the case that, for every $\epsilon>0$,

• there is some $3$-sleeping bag with volume $\le v+\epsilon$ which does contain a $2$-sleeping bag as a cross section? (EDIT: As Will Sawin points out below, in order to avoid a trivial answer to this question we need to further require that the $3$-sleeping bag not have any subset which is also a $3$-sleeping bag but does not have any $2$-sleeping bag cross-sections.)

• there is some $3$-sleeping bag with volume $\le v+\epsilon$ which does not contain a $2$-sleeping bag as a cross section?