there are some examples in convex analysis. For example, in the book of Borwein & Lewis (Convex Analysis and nonlinear optimization) page 225 they use the PVE to prove that every Chebyshev set (i.e. every set that has the property "every point has a unique nearest point") is in fact convex (in finite dimensional spaces). The problem in arbitrary Banach spaces is open (as far I know). In general, I think that if you can to study some property of a set and you are able to associate a lsc function bounded from below function then you can use the PVE to get some properties of the set. This is specially useful because this framework is more adequate to work with nonsmooth function. In fact, you can take the distance function to a set and define some class of set saying some property of this function. This has been done to define, for example, the class of regular sets, prox-regular sets, etc.

There are some examples in convex analysis. For example, in the book of Borwein & Lewis (Convex Analysis and Nonlinear Optimization) page 225, they use the PVE to prove that every Chebyshev set (i.e. every set that has the property "every point has a unique nearest point") is in fact convex (in finite-dimensional spaces). The problem in arbitrary Banach spaces is open (as far I know).

In general, I think that if you can study some property of a set and you are able to associate a lsc function which is bounded from below, then you can use the PVE to get some properties of the set. This is specially useful because this framework is more adequate to work with nonsmooth functions. In fact, you can take the distance function to a set and define some class of set saying some property of this function. This has been done to define, for example, the class of regular sets, prox-regular sets, etc.