The least informative (favorable) distributions, minimizing Fisher information for a multivariate location parameter, are derived in the parametric class of the exponential-power spherically symmetric distributions under the following characterizing restrictions; (i) a bounded variance, (ii) a bounded value of a density at the center of symmetry, and (iii) the intersection of these restrictions. In the first two cases, (i) and (ii) respectively, the least informative distributions are the Gaussian and Laplace, respectively. In the latter case (iii) the optimal solution has three branches, with relatively small variances it is the Gaussian, with relatively large variances it is the Laplace, and it is the compromise between them with intermediate variances. The corresponding robust minimax M-estimators of location are given by the L_2-norm, the L_1-norm and the L_p-norm methods. The properties of the proposed estimators and their adaptive versions are studied in asymptotics and on finite samples by Monte Carlo.