  In this paper we consider under-determined systems of linear equations that
have sparse solutions. This subject attracted enormous amount of interest in
recent years primarily due to influential works \cite{CRT,DonohoPol}. In a
statistical context it was rigorously established for the first time in
\cite{CRT,DonohoPol} that if the number of equations is smaller than but still
linearly proportional to the number of unknowns then a sparse vector of
sparsity also linearly proportional to the number of unknowns can be recovered
through a polynomial $\ell_1$-optimization algorithm (of course, this assuming
that such a sparse solution vector exists). Moreover, the geometric approach of
\cite{DonohoPol} produced the exact values for the proportionalities in
question. In our recent work \cite{StojnicCSetam09} we introduced an
alternative statistical approach that produced attainable values of the
proportionalities. Those happened to be in an excellent numerical agreement
with the ones of \cite{DonohoPol}. In this paper we give a rigorous analytical
confirmation that the results of \cite{StojnicCSetam09} indeed match those from
\cite{DonohoPol}.
