In this paper we revisit the exsistence theorem for $L^r$-optimal quantization, $r\ge 2$, with respect to a Bregman divergence: we establish the existence of optimal quantizaers under lighter assumptions onthe strictly convex function which generates the divergence, espcially in the quadratic case ($r=2$). We then prove a uniqueness theorem ``à la Trushkin'' in one dimension for strongly unimodal distributions and divergences gerated by strictly convex functions whiose thire dervative is either stictly $\log$-convex or $\log$-concave.