The generalized spectral radius ( ) of a set of n n matrices is ( ) = lim sup k → ∞ k( ) 1 k , where k ( ) = sup{ρ(A 1 A 2 A k ): each A i }. Thejoint spectral radius ( ) is circ;( ) = lim sup k → ∞ circ; k ( ) 1 k , where circ;( ) = sup{ A 1 A k : eachA i }. It is known that ( ) = ( ) holds for any finite set of n n matrices. The finiteness conjecture asserts that for any finite set of real n n matrices there exists a finite k such that circ;( ) = ( ) 1 k . The normed finiteness conjecture for a given operator norm asserts that for any finite set = {A 1 , ,A m } having all A i o p ≤ 1, either circ;( ) < 1 orcirc; ( ) = ( ) = k ( ) 1 k = 1 for some finitek . It is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms. The normed finiteness conjecture is proved for a large class of operator norms, extending results of Gurvits. In particular, for polytope norms and for the Euclidean norm, explicit upper bounds are given for the least k having ( ) = k ( ) 1 k . These results imply upper bounds for generalized critical exponents for these norms.