This is an announcement for the paper "Multi-norms and the injectivity of $L^p(G)$" by H. Garth Dales, Matthew Daws, Hung Le Pham, Paul Ramsden. Abstract: Let $G$ be a locally compact group, and take $p\in(1,\infty)$. We prove that the Banach left $L^1(G)$-module $L^p(G)$ is injective (if and) only if the group $G$ is amenable. Our proof uses the notion of multi-norms. We also develop the theory of multi-normed spaces. Archive classification: math.FA Mathematics Subject Classification: 46H25, 43A20 Remarks: 27 pages Submitted from: matt.daws@cantab.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4320 or http://arXiv.org/abs/1101.4320