On Two Information Quantities Relating Two Distortion Balls

This paper clarifies the relationship between two information quantities related to two distortion balls in variable-length lossy source coding. To show various fundamental limits in variable-length lossy source coding, the notion of distortion ball has been known to be useful. In the previous study by Kostina et al., it was shown that the fundamental limit of the minimum average codeword lengths under an excess distortion constraint is characterized by the information quantity related to the distortion ball centered at a source symbol. On the other hand, in our previous study, it was shown that the same fundamental limit is characterized by the information quantity related to the distortion ball centered at a reproduction symbol. Then, what is the relationship between these two information quantities? This paper gives an answer to this question.