This also works:

pts = {{1, 1}, {-1, 1}, {-1, 0}, {-1/2, 0},
   {0, -1/2}, {0, -1}, {1, -1}};
RegionDifference[Rectangle[{-2, -2}, {2, 2}], Polygon[pts]]

Thanks for the different approaches — this clarifies the distinction between bounding with a finite rectangle and using FullRegion.

I have a related question: if the goal is to work symbolically with the true exterior region (unbounded) — for example to integrate over it or apply further region operations — is RegionDifference[FullRegion[2], [\[Polygon\]][1] ][pts]] always the preferred formulation?

In particular:

• Does using FullRegion[2] have any performance or simplification implications compared to a large bounding rectangle?
• Are there cases where Reduce[!Element[{x,y}, poly], …] would produce a more suitable representation for downstream symbolic manipulation?

I’m trying to understand which representation is more robust when the exterior region is used in subsequent analytic computations rather than just plotting.

Appreciate any insight.

Your method has the boundary region going from -2 to 2 in the x and y directions. My method has the boundary region going from -Infinity to Infinity in the x and y directions.