Characterization of a generalized triangle inequality in normed spaces

For a normed linear space (X,‖⋅‖)(X,‖⋅‖) and p>0p>0 we characterize all nn-tuples (μ1,…,μn)∈Rn(μ1,…,μn)∈Rn for which the generalized triangle inequality of the second type ‖x1+⋯+xn‖p≤‖x1‖pμ1+⋯+‖xn‖pμn holds for any x1,…,xn∈Xx1,…,xn∈X. We also characterize (μ1,…,μn)∈Rn(μ1,…,μn)∈Rn for which the reverse of the inequality above holds.