The function log log 1/(1 - e^x) = log (\sum_{k > 0} e^{kx}/k) is convex: it is a log-sum-exp with an infinite sum, in its natural domain x < 0, i.e. where the inner logarithm can be expressed as a power series.
Alternatively, instead of expressing this function f(x), it is possible to ask to express its epigraph f(x) <= t as a convex set in cvx rules.
I would prefer to avoid the infinite sum because of convergence issues. Is there any other way to express this problem in a canonical form? I tried various substitutions for the exponential cone, but no success.

This is essential for one combinatorial construction in our paper

Here is a somewhat similar, although concave expression, which no one of the forum has really figured out how to do. Log(log(1+exp(x)) objective function

Concerning the function presented by the link. The inverse function mentioned there, log(exp(exp(x)) - 1) is also of interest in our paper, it corresponds to another combinatorial construction that we would like to cover, i have no ideas about it neither.

The function log(exp(exp(x)) - 1) was already considered in this post, where (6e) is rewritten to (11) which is log-sum-exp + this function. Our conclusion in post #12 has unfortunately not changed, but we do try to keep track of the open questions.

can be rewritten as x \leq \log\left(1 - \frac{1}{\exp(\exp(t))}\right)

which is equivalent to x \leq \log\left(1 - \frac{1}{r}\right),\quad r = \exp(\exp(t))

where the former is just a simple set. Unfortunately, the latter cannot be relaxed to r \geq \exp(\exp(t)) as r would just fly off to infinity relaxing the implications on x â€“ i.e., the composition rules are violated

This may be another one of those sets I donâ€™t know how to representâ€¦

Ok, in your last attempt you are substituting a convex function \exp(\exp(t)) > 1 into a concave increasing \log(1 - 1/r) which is not allowed by cvx. Alternatively, in the initial formulation it is also impossible because the resulting function cannot be represented as a composition of convex ones.

Well, I give up at this point. I personally believe that these two functions cannot be expressed using the exponential cones. But I am not sure that the rigorous proof of this statement can be useful somewhere. As you guys have said in a different thread somewhere, let us stop chasing unicorns and letâ€™s try to go write some self-concordant barriers

According to my findings, a usual logarithmic barrier with self-concordance parameter m will cover the part x > -C m, (C around log 10 natural base?) this is just an empirical observation combined with some lemmas from Nesterov/Nemirovski. Using some additional knowledge about my specific problem I conclude that if I take m large enough, this is sufficient for me, but in general, I donâ€™t know how to construct a self-concordant barrier for this one, covering all the ray R^{-}. I suspect that for the inverse function of log(exp(exp(x))-1) the situation should be similar.

Following the discussion: we have uploaded a paper on arxiv that discusses the two functions (as a part of restricted CYC and SET combinatorial constructions).

I only give empirical treatment and justify it in the necessary range. I suspect that the second one, log exp exp -1 has a self-concordant logarithmic barrier in the full range (but we donâ€™t give the proof of that). This information can be however useful to others.