  This paper considers compressed sensing and affine rank minimization in both
noiseless and noisy cases and establishes sharp restricted isometry conditions
for sparse signal and low-rank matrix recovery. The analysis relies on a key
technical tool which represents points in a polytope by convex combinations of
sparse vectors. The technique is elementary while leads to sharp results.
  It is shown that for any given constant $t\ge {4/3}$, in compressed sensing
$\delta_{tk}^A < \sqrt{(t-1)/t}$ guarantees the exact recovery of all $k$
sparse signals in the noiseless case through the constrained $\ell_1$
minimization, and similarly in affine rank minimization
$\delta_{tr}^\mathcal{M}< \sqrt{(t-1)/t}$ ensures the exact reconstruction of
all matrices with rank at most $r$ in the noiseless case via the constrained
nuclear norm minimization. Moreover, for any $\epsilon>0$,
$\delta_{tk}^A<\sqrt{\frac{t-1}{t}}+\epsilon$ is not sufficient to guarantee
the exact recovery of all $k$-sparse signals for large $k$. Similar result also
holds for matrix recovery. In addition, the conditions $\delta_{tk}^A <
\sqrt{(t-1)/t}$ and $\delta_{tr}^\mathcal{M}< \sqrt{(t-1)/t}$ are also shown to
be sufficient respectively for stable recovery of approximately sparse signals
and low-rank matrices in the noisy case.
