  Compressed sensing deals with the recovery of sparse signals from linear
measurements. Without any additional information, it is possible to recover an
$s$-sparse signal using $m \gtrsim s \log(d/s)$ measurements in a robust and
stable way. Some applications provide additional information, such as on the
location of the support of the signal. Using this information, it is
conceivable the threshold amount of measurements can be lowered. A proposed
algorithm for this task is \emph{weighted $\ell_1$-minimization}. Put shortly,
one modifies standard $\ell_1$-minimization by assigning different weights to
different parts of the index set $[1, \dots d]$. The task of choosing the
weights is however non-trivial.
  This paper provides a complete answer to the question of an optimal choice of
the weights. In fact, it is shown that it is possible to directly calculate
unique weights that are optimal in the sense that the threshold amount of
measurements needed for exact recovery is minimized. The proof uses recent
results about the connection between convex geometry and compressed
sensing-type algorithms.
