  In this paper we investigate the convexity of the relevance-compression
function for the Information Bottleneck and the Information Distortion
problems. This curve is an analog of the rate-distortion curve, which is
convex. In the problems we discuss in this paper, the distortion function is
not a linear function of the quantizer, and the relevance-compression function
is not necessarily convex (concave), but can change its convexity. We relate
this phenomena with existence of first order phase transitions in the
corresponding Lagrangian as a function of the annealing parameter.
