  We are concerned with geometric properties of transonic shocks as free
boundaries in two-dimensional self-similar coordinates for compressible fluid
flows, which are not only important for the understanding of geometric
structure and stability of fluid motions in continuum mechanics but also
fundamental in the mathematical theory of multidimensional conservation laws. A
transonic shock for the Euler equations for self-similar potential flow
separates elliptic (subsonic) and hyperbolic (supersonic) phases of the
self-similar solution of the corresponding nonlinear partial differential
equation in a domain under consideration, in which the location of the
transonic shock is apriori unknown. We first develop a general framework under
which self-similar transonic shocks, as free boundaries, are proved to be
uniformly convex, and then apply this framework to prove the uniform convexity
of transonic shocks in the two longstanding fundamental shock problems -- the
shock reflection-diffraction by wedges and the Prandtl-Meyer reflection for
supersonic flows past solid ramps. To achieve this, our approach is to exploit
underlying nonlocal properties of the solution and the free boundary for the
potential flow equation.
