(joint with Johan Håstad och Johan Wästlund)

By a \emph{sleeping bag} for a baby snake in $d$ dimensions we
mean a subset of $\mathbb{R}^d$ which can cover, by rotation and
translation, every curve of unit length. We construct sleeping
bags which are smaller than any previously known in dimensions 3
and higher. In particular, we construct a three dimensional
sleeping bag of volume approximately 0.075803. For large $d$ we
construct $d$-dimensional sleeping bags with volume less than
$\frac{(c\sqrt{\log d})^d}{d^{3d/2}}$ for some constant $c$.

To obtain the last result, we show that every curve of unit length
in $\mathbb{R}^d$ lies between two parallel hyperplanes at
distance at most $c_{1}d^{-3/2}\sqrt{\log d}$, for some constant
$c_{1}$.