  $L_p$-Christoffel-Minkowski problem arises naturally in the
$L_p$-Brunn-Minkowski theory. It connects both curvature measures and area
measures of convex bodies and is a fundamental problem in convex geometric
analysis. Since the lack of Firey's extension of Brunn-Minkowski inequality and
constant rank theorem for $p<1$, the existence and uniqueness of
$L_p$-Brunn-Minkowski problem are difficult problems. In this paper, we prove a
uniqueness theorem for solutions to $L_p$-Christoffel-Minkowski problem with
$p<1$ and constant prescribed data. Our proof is motivated by the idea of
Brendle-Choi-Daskaspoulos's work on asymptotic behavior of flows by powers of
the Gaussian curvature. One of the highlights of our arguments is that we
introduce a new auxiliary function $Z$ which is the key to our proof.
