This paper proposes a method for finding the maximal mechanical power, that can be developed for a cyclist without excess of carbon dioxide production. The method is based on a recently proposed discrete-time linear parameter-varying model of gas exchange dynamics. In addition, the problem is solved by using an ellipsoidal invariant set which characterizes all possible trajectories of the system state in presence of a bounded pedal power. The latter allows us to tackle the problem as a geometrical optimization problem involving an inequality constraint. The numerical example concerns two scenarios, which cover moderate and high intensity exercises.