  We develop a new approach to vector quantization, which guarantees an
intrinsic stationarity property that also holds, in contrast to regular
quantization, for non-optimal quantization grids. This goal is achieved by
replacing the usual nearest neighbor projection operator for Voronoi
quantization by a random splitting operator, which maps the random source to
the vertices of a triangle of $d$-simplex. In the quadratic Euclidean case, it
is shown that these triangles or $d$-simplices make up a Delaunay triangulation
of the underlying grid. Furthermore, we prove the existence of an optimal grid
for this Delaunay -- or dual -- quantization procedure. We also provide a
stochastic optimization method to compute such optimal grids, here for higher
dimensional uniform and normal distributions. A crucial feature of this new
approach is the fact that it automatically leads to a second order quadrature
formula for computing expectations, regardless of the optimality of the
underlying grid.
