A Smaller Sleeping Bag for a Baby Snake

By a sleeping bag for a baby snake in d dimensions we mean a subset of Rd 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 (c √ log d)d/d3d/2 for some constant c. To obtain the last result, we show that every curve of unit length in Rd lies between two parallel hyperplanes at distance at most c1d−3/2 √ log d, for some constant c1.