Convexity of shock polars

We show that the shock polars of compressible full potential flow are strictly convex if the enthalpy per mass is a convex function of volume per mass, in particular when the sound speed is a nondecreasing function of density. Counterexamples are given for some cases that violate the enthalpy condition. For the full Euler equations with convex equation of state satisfying the ideal gas law, polars are strictly convex if heat capacity is constant, but counterexamples are given in variable cases, showing no useful generalizations are possible.