This is an announcement for the paper "The Gelfand widths of $\ell_p$-balls for $0<p\leq 1$" by Simon Foucart, Alain Pajor, Holger Rauhut, and Tino Ullrich. Abstract: We provide sharp lower and upper bounds for the Gelfand widths of $\ell_p$-balls in the $N$-dimensional $\ell_q^N$-space for $0<p\leq 1$ and $p<q \leq 2$. Such estimates are highly relevant to the novel theory of compressive sensing, and our proofs rely on methods from this area. Archive classification: math.FA cs.IT math.IT Mathematics Subject Classification: 41A46, 46B09 Remarks: 15 pages The source file(s), GelfandSAHTarxiv.tex: 45830 bytes, is(are) stored in gzipped form as 1002.0672.gz with size 15kb. The corresponding postcript file has gzipped size 84kb. Submitted from: tino.ullrich@hcm.uni-bonn.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1002.0672 or http://arXiv.org/abs/1002.0672 or by email in unzipped form by transmitting an empty message with subject line uget 1002.0672 or in gzipped form by using subject line get 1002.0672 to: math@arXiv.org.