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

Cordially