Information theory and convex optimization
The books below may be more to your liking, but in general, the texts/lecture notes are written for the use of (mainly) postgraduate students in engineering and cannot presume deep knowledge of convex analysis. 


Csizsar, I., and Korner, J., Information Theory: Coding Theorems for Discrete Memoryless Systems, 2nd Ed, Cambridge.
Berger, T., Rate Distortion Theory, quite old probably late 70's or early 80s, can't remember the publisher maybe Wiley.
Yeung, R.W., A First Course in Information Theory, Springer.


The research articles on Shannon theory and related fields in, say, IEEE Transactions on Information Theory, may fit the bill better, though not always. 

An older text which may also be of interest is 

Wolfowitz, J., Coding Theorems of Information Theory, Springer, 1960's.