  In this paper, for a Borel probability measure $P$ on a normed space $\mathbb
R^k$, we extend the definitions of $n$th unconstrained quantization error,
unconstrained quantization dimension, and unconstrained quantization
coefficient, which traditionally in the literature are known as $n$th
quantization error, quantization dimension, and quantization coefficient, to
the definitions of $n$th constrained quantization error, constrained
quantization dimension, and constrained quantization coefficient. The work in
this paper extends the theory of quantization and opens a new area of research.
In unconstrained quantization, the elements in an optimal set are the
conditional expectations in their own Voronoi regions, and it is not true in
constrained quantization. In unconstrained quantization, if the support of $P$
contains infinitely many elements, then an optimal set of $n$-means always
contains exactly $n$ elements, and it is not true in constrained quantization.
It is known that the unconstrained quantization dimension for an absolutely
continuous probability measure equals the Euclidean dimension of the underlying
space. In this paper, we show that this fact is not true as well for the
constrained quantization dimension. It is known that the unconstrained
quantization coefficient for an absolutely continuous probability measure
exists as a unique finite positive number. From work in this paper, it can be
seen that the constrained quantization coefficient for an absolutely continuous
probability measure can be any nonnegative number depending on the constraint
that occurs in the definition of $n$th constrained quantization error.
