Optimal quantization (Q1811226)

L. Fejes Tóth's inequality for the sums of moments [\textit{L. Fejes Tóth}, Am. J. Math. 70, 174--180 (1948; Zbl 0038.35603)] is one of the basic inequalities in discrete geometry. A stability version [Geom. Dedicata 84, 271--320 (2001; Zbl 0982.52020)] was obtained by \textit{P. Gruber}. In the survey under review these results and their extensions to dimensions \(\geq 3\) are described in detail. Important applications of these results in the following areas are discussed: distortion of vector quantizers, approximation of probability measures, numerical integration, approximation of convex bodies and isoperimetric problems in normed spaces.