Thank you, did you have a chance to look at these sources? Unfortunately, I can't find any preview online and my university library doesn't have them as well.

Thank you, Sergei! There is a specific reason, why the $L_1$-balls are important, rather than $L_\infty$-balls. However, because I am ultimately interested in some specific class of linear mappings, the combinatorial type is fixed and one can get a closed formula.

Thank you, but I am not sure I understand how can this be used to compute the $\mathcal{L}^m(f(S))$, which I am interested in. Also, shouldn't the formula have $\mathcal{H}^{n - m}$, rather than $\mathcal{H}^{m - n}$ on the left-hand side?

Do you mean simple cycle?

Boolean circuit is an acyclic graph, are you sure that accessibility problem for acyclic graphs is still $NLOGSPACE$-complete?

You can modify it like this: keep the set of numbers $S$ and then $n - 1$ times extract two minimal elements from the set, calculate their $lcm$ and then put it back into the set.

Well, the big-O notation gives us an upper bound on the complexity, so the bounds I gave hold, but probably can be further improved.