Uniqueness of locally optimal quantizer for log-concave density and convex error weighting function

It is desired to encode a random variable
X
using an
N
-level quantizer
Q
to minimize the expected distortion
E \rho(|X-Q(X))I)
, where the error weighting function
\rho
is convex, strictly increasing and continuously differentiable. It is shown that if
X
has a log-concave density, then there exists a unique locally optimal quantizer
Q \ast
and Lloyd's Method I may be used to find
Q \ast
. Trushkin had earlier shown this result for the error weighting functions
\rho (t) \equiv t
and
\rho(t) \euiv t^{2}
.