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We say a map f:X \to Y is an \epsilon-embedding if it is continuous and thediameter of the fibres is less than \epsilon. This type of maps is used in thenotion of Urysohn width (sometimes referred to as Alexandrov width), a_n(X). Itis the smallest real number such that there exists an \epsilon-embedding from Xto a n-dimensional polyhedron. Surprisingly few estimations of these numberscan be found, and one of the aims of this paper is to present some. FollowingGromov, we take the slightly different point of view by looking at the smallestdimension n for which there exists a \epsilon-embedding to a polyhedron ofdimension n. While bounds are obtained using Hadamard matrices, the Borsuk-Ulamtheorem, the filling radius of spheres, and lower bounds for the diameter ofsets of n+1 points not contained in a hemisphere (obtained by methods veryclose to those of Ivanov and Pichugov). We are also able to give a completedescription in dimension 3 for 1 \leq p \leq 2.

Width of l^p balls