Asymptotic Optimality of Scalar Gersho Quantizers

In his famous paper (Gersho, IEEE Trans. Inf. Theory 25(4):373–380, 1979), Gersho stressed that the codecells of optimal quantizers asymptotically make an equal contribution to the distortion of the quantizer. Motivated by this fact, we investigate in this paper quantizers in the scalar case, where each codecell contributes with exactly the same portion to the quantization error. We show that such quantizers of Gersho type—or Gersho quantizers for short—exist for nonatomic scalar distributions. As a main result, we prove that Gersho quantizers are asymptotically optimal.