Please be more specific and write out the constraints defining the set C.
The crucial thing is not whether the author of a book or paper uses set membership to denote constraints, rather than writing them out, it is what the constraints are. So the similarity to the link to the book may be superficial and irrelevant.
Given that you mention quantum entropies and Tr(X logX), it sounds like CVXQUAD https://github.com/hfawzi/cvxquad might suit your needs. it adds several functions to CVX, including
quantum_entr(X) = -trace(X*logm(X)) , concave in X
So you could have quantum_entr(X) >= v
But your constraint is -quantum_entr(X) >= v, which is a non-convex constraint. So hopefully you just have a typo in your post.
There are other functions such as
quantum_rel_entr(X,Y) = trace(X*(logm(X)-logm(Y))) ,convex in (X,Y) . Beware though, the dimensions of the (several) matrices constrained to be semidefinite for quantum_rel_entr are each of the order 2n^2 by 2n^2 when X and Y are n by n; whereas the several are “only” of order 2n by 2n for quantum_entr. See the table on the top of p. 3 of https://arxiv.org/pdf/1705.00812.pdf
See my post Perspective of log det function, and CVX formulations of related log det problems using Quantum Relative Entropy from CVXQUAD for examples of how to formulate and solve various matrix perspective and log_det related functions using CVXQUAD in CVX.
You may also be interested in the latter sections of CVXQUAD: How to use CVXQUAD's Pade Approximant instead of CVX's unreliable Successive Approximation for GP mode, log, exp, entr, rel_entr, kl_div, log_det, det_rootn, exponential cone. CVXQUAD's Quantum (Matrix) Entropy & Matrix Log related functions . Note, there is no need for you to use CVXQUAD’s exponential.m replacement if you are trying to use the “quantum” related functions in CVXQUAD.