Approximation of diameters: Randomization doesn't help

We describe a deterministic polynomial-time algorithm which, for a convex body K in Euclidean n-space, finds upper and lower bounds on K's diameter which differ by a factor of O(√n/log n). We show that this is, within a constant factor, the best approximation to the diameter that a polynomial-time algorithm can produce even if randomization is allowed. We also show that the above results hold for other quantities similar to the diameter - namely, inradius, circumradius, width, and maximization of the norm over K. In addition to these results for Euclidean spaces, we give tight results for the error of deterministic polynomial-time approximations of radii and norm-maxima for convex bodies in finite-dimensional l_p spaces.