Is it possible for a reversible path between two states to be both adiabatic and isothermal, for a system having any number of independent variables?
Yes, for example, mixture of liquid and solid water at 0 Celsius and 1 bar in an insulated container, getting slowly compressed by a piston with infinitesimal overpressure. As the piston moves down, ice melts, work is done on the system, and the system temperature remains at 0 Celsius until all ice melts.
Another question: If I have two states in an n dimensional space which contains all possible states of a system with fixed composition, is it always possible to go from one state to the other state by either a reversible adiabatic change or a reversible isothermal change or a combination of adiabatic and isothermal reversible changes? Why or why not?
Not by an adiabatic path alone, because that is isentropic, it doesn't change entropy of the system, thus it can only end up in a state which has the same entropy.
Similar with an isothermal path, the end state has the same temperature.
Combination of both types can access much greater set of states than the single type alone, and for simple systems such as single gas, it can reach all of its possible equilibrium states. But I don't know if it can reach all of the possible states in general, for more complicated thermodynamic systems with more state variables than two. It seems unlikely.
