A Smaller Sleeping Bag for a Baby Snake

By a sleeping bag for a baby snake in d dimensions we mean a subset of 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 \(\left( {c\sqrt {\log d} } \right)^d /d^{3d/2}\) for some constant c.

To obtain the last result, we show that every curve of unit length in R ^d lies between two parallel hyperplanes at distance at most \(c_1 d^{ - 3/2} \sqrt {\log d}\), for some constant c ₁ .

Authors and Affiliations

1. NADA, Royal Institute of Technology, S-100 44, Stockholm, Sweden

   J. Håstad

2. Department of Mathematics, Linköping University, S-581 83, Linköping, Sweden

   S. Linusson & J. Wästlund

Received March 24, 1999, and in revised form July 26, 2000. Online publication April 6, 2001.

An erratum to this article is available at http://dx.doi.org/10.1007/s00454-007-0381-4.

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