what about if we replace arbitrary $r$ with $r=lcm (m,n)$? namely for every $m,n$ there are elements $g,h$ with $|g|=m,\quad |h|=n,\quad |gh|=\frac{mn}{gcd(m,n)}$?

@GilesGardam What about a possible "simple" example: a simple group with the property under discussion?The example you mentioned is a direct product of all possible finite group mentioned in theorem 1.64(for all m,n, r) but it is not a simple group. So is there a simple example and then classification of all simple example would be the next question

@DanielAsimov BTW may I ask you to look at the updated question (a relation to Arnold conjecture)?

@DanielAsimov Yes Perfect thank you

@DanielAsimov The tragedy is that the Poincare Birkhoff is about topological $[0 1]\times S^1$ but it is not about product metric

@DanielAsimov What is a reference for the Moser theorem you mentioned?

@DanielAsimov Perhaps one possible sense could be the following: We change the metric on $\mathbb{R}^n$ and compute the volum wrt the new metric. The convexity can have two interpretation: wrt the straight lines or wrt the geodesics of metrics. Is it a reasonable sense for consideration of the generalized Fenchel theorem?

So if the multiplication version of the theorem is true we can not conclude that the Poincare Birkhoff theorem is true for arbitrary volum form. Am I mistaken?

@DanielAsimov I correct a typos: $\int_M f^* \alpha =\int_M \alpha$

@DanielAsimov Is it realy the case? if it would be true so one may associate an intrinsic volum to every manifold M: the integral of arbitrary volum form over M. By theorem of change of variabble $\int_M f^* \alpha =\int M \alpha$. But I guess the theorem of Moser has a function multiplication too that is $\alpha =g f^* \beta$ for a positive function $g$. Do you agree?

@MartinM.W. Thank you! could you please give me a reference. I googled "Rotation set" and Lie group simultaneously but I could not find materials in this regard