Sharp constants related to the triangle inequality in Lorentz spaces

We study the Lorentz spaces $L^{p,s}(R,\mu)$ in the range $1<p<s\le \infty$, for which the standard functional $$||f||_{p,s}=(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t})^{1/s}$$ is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: $$||f||_{(p,s)}=\inf\bigg\{\sum_{k}||f_k||_{p,s}\bigg\},$$ where the infimum is taken over all finite representations $f=\sum_{k}f_k.$ We also prove that the decomposition norm and the dual norm $$||f||_{p,s}'= \sup\left\{\int_R fg d\mu: ||g||_{p',s'}=1\right\}$$ agree for all values $p,s>1$.