To characterize the geometry of a measure, its so-called generalized dimensions D(q) have been introduced recently. The mathematically precise definition given by Falconer turns out to be unsatisfactory for reasons of convergence as well as of undesired sensitivity to the particular choice of coordinates in the negative q range. A new definition is introduced, which is based on box-counting too, but which carries relevant information also for negative q. In particular, rigorous proofs are provided for the Legendre connection between generalized dimensions and the so-called multifractal spectrum and for the implicit formula giving the generalized dimensions of self-similar measures, which was until now known only for positive q.