Let V be a real or complex finite-dimensional vector space, and let be a set of norms on V. The norm hull of a vector x ∈ V with respect to is the set of vectors y ∈ V that satisfy ∥y∥ ≤ ∥x∥ for all ∥ · ∥ ∈ N. We study and give characterization of the norm hull for some sets of well-known norms on general vector spaces, and for the set of algebra norms and the set of induced norms on the algebra of n × n real or complex matrices