  We consider the optimization of a finite-time Carnot engine characterized by
small dissipations. We bound the power with a simple inequality and show that
the optimal strategy is to perform small cycles around a given working point,
which can be thus chosen optimally. Remarkably, this optimal point is
independent of the figure of merit combining power and efficiency that is being
maximized. Furthermore, for a general class of dynamics the power output
becomes proportional to the heat capacity of the working substance. Since the
heat capacity can scale supra-extensively with the number of constituents of
the engine, this enables us to design optimal many-body Carnot engines reaching
maximum efficiency at finite power per constituent in the thermodynamic limit.
