This is an announcement for the paper "Sharp constants related to the triangle inequality in Lorentz spaces" by Sorina Barza, Viktor Kolyada, and Javier Soria. Abstract: We study the Lorentz spaces $L^{p,s}(R,\mu)$ in the range $1<p<s\le \infty$, for which the standard functional $$ ||f||_{p,s}=\left(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t}\right)^{1/s} $$ 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 $p,s>1$. Archive classification: math.FA math.CA Mathematics Subject Classification: 46E30, 46B25 Remarks: 24 pages The source file(s), Norms-Constants.tex: 47398 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.0647 or http://arXiv.org/abs/0709.0647 or by email in unzipped form by transmitting an empty message with subject line uget 0709.0647 or in gzipped form by using subject line get 0709.0647 to: math@arXiv.org.