  The literature on sparse recovery often adopts the $l_p$ "norm" $(p\in[0,1])$
as the penalty to induce sparsity of the signal satisfying an underdetermined
linear system. The performance of the corresponding $l_p$ minimization problem
can be characterized by its null space constant. In spite of the NP-hardness of
computing the constant, its properties can still help in illustrating the
performance of $l_p$ minimization. In this letter, we show the strict increase
of the null space constant in the sparsity level $k$ and its continuity in the
exponent $p$. We also indicate that the constant is strictly increasing in $p$
with probability $1$ when the sensing matrix ${\bf A}$ is randomly generated.
Finally, we show how these properties can help in demonstrating the performance
of $l_p$ minimization, mainly in the relationship between the the exponent $p$
and the sparsity level $k$.
