Optimum quantization and its applications

Minimum sums of moments or, equivalently, distortion of optimum quantizers play an important role in several branches of mathematics. Fejes Tóth's inequality for sums of moments in the plane and Zador's asymptotic formula for minimum distortion in Euclidean d-space are the first precise pertinent results in dimension d?2. In this article these results are generalized in the form of asymptotic formulae for minimum sums of moments, resp. distortion of optimum quantizers on Riemannian d-manifolds and normed d-spaces. In addition, we provide geometric and analytic information on the structure of optimum configurations. Our results are then used to obtain information on

(i)

the minimum distortion of high-resolution vector quantization and optimum quantizers,

(ii)

the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures,

(iii)

the minimum error of numerical integration formulae for classes of Hölder continuous functions and optimum sets of nodes,

(iv)

best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes, and

(v)

the minimum isoperimetric quotient of convex polytopes in Minkowski spaces and the form of the minimizing polytopes.