Let X be a separable, infinite-dimensional Banach space and N(X) the set of equivalent norms in X. An element x of X is said to be a "co-smooth" direction of a morm, if that norm is Gateaux-smooth everywhere in the direction of x. Let G(0) be the set of norms admitting at least one co-smooth direction x (not 0). What is the complexity of G(0)? Counting quantifiers leads to the class PCA, but this is too large. The set G(0) can be obtained by applying the Hausdorff operation A to a scheme of co-analytic sets. Moreover, even when X is not too exotic, the class G(0) reduces every set of the specified class by a continuous map into N(X).