The Gelfand widths of ${\mathrm{\ell }}_p$-balls for $0<p\le 1$

We provide sharp lower and upper bounds for the Gelfand widths of ${\mathrm{\ell }}_p$-balls in the $N$-dimensional ${\mathrm{\ell }}_q^N$-space for $0<p\le 1$ 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.