# Small ball probablities and the quantization problem for Gaussian measures.

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Let $\mu$ be a probability measure on a metric space $(E,d)$ and $N$ a positive integer.

The {\em quantization error} $e_N$ of $\mu$ is defined as the infimum over all subsets ${\cal{E}}$

of $E$ of cardinality $N$ of the average distance w.~r.~t.~$\mu$ to the closest point in the set

${\cal{E}}$. We study the asymptotics of $e_N$ for large $N$. We concentrate on the

case of a Gaussian measure $\mu$ on a Banach space. The asymptotics of $e_N$ is closely related to

{\em small ball probabilities} which have received considerable interest in the past decade.

The quantization problem is motivated by the problem of encoding a continuous signal

by a specified number of bits with minimal distortion. This is joint work with Steffen Dereich,

Franz Fehringer, Anis Matoussi and Michail Lifschitz.