Optimal quantizers for Radon random vectors in a Banach space

For n∈N,r∈(0,∞) and a Radon random vector X with values in a Banach space E let , where the infimum is taken over all subsets α of E with card(α)⩽n (n-quantizers). We investigate the existence of optimal n-quantizers for this Lr-quantization problem, derive their stationarity properties and establish for Lp-spaces E the pathwise regularity of stationary quantizers.