Tel Aviv University

We compute the number of summands in q-averages of norms needed to approximate an Euclidean norm. It turns out that these numbers depend on the norm involved essentially only through the maximal ratio of the norm and the Euclidean norm. Particular atten-tion is given to the case q = ∞ (in which the average is replaced with the maxima). This is closely connected with the behavior of certain families of projective caps on the sphere. 1. Introduction. The starting point of this paper is a result from [MS1] which we would like to recall. Denote by | · | the canonical Euclidean norm on IRn. Given another norm ‖ · ‖ on IRn denote by a and b the smallest constants such that (1.1) a−1|x | ≤ ‖x ‖ ≤ b|x|, for all x ∈ IRn