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An example can be found here.

An example can be found here.

Edit:

SimplePolygonPartition can be used to divide self-intersecting Polygon to simple pieces. The usage is like

SimplePolygonPartition[Polygon[...]] SimplePolygonPartition[Polygon[...],Graphics`Region`RegionDump`FillingMethod->"OddEvenRule"] 

An example can be found here.

AppendTo[$ContextPath, "Region`"] Clear[testfunc] testfunc[reg_] := {ToString /@ #, Through[#[reg]]} &[{ ConvexRegionQ, BoundedRegionQ, Module[{dim = RegionDimension[#]}, var = Symbol["x" <> ToString[#]] & /@ Range[dim];  dim] &,  RegionEmbeddingDimension, RegionMeasure, RegionCentroid, RegionProperty[#, var, "FastDescription"] &, RegionProperty[#, var, "ImplicitDescription"] &, RegionElement, LevelFunction[RegionProperty[#, var, "FastDescription"][[1, 2]]] & }] // Grid[Insert[#, {ConvexRegionQ, BoundedRegionQ, RegionDimension, RegionEmbeddingDimension, RegionMeasure, RegionCentroid, FastDescription, ImplicitDescription, RegionElement, LevelFunction}, 2]\[Transpose], Dividers -> All, FrameStyle -> GrayLevel[.8], Alignment -> Left] & // Quiet 

In addition of BoxRegion, other *Regions also seems to be used to declare regions:

Maybe some of them (FastDescription, ImplicitDescription, LevelFunction) work only on "full-rank" regions?

Some regions look very special (or should I say general?):

AppendTo[$ContextPath, "Region`"] Clear[testfunc] testfunc[reg_] := {ToString /@ #, Through[#[reg]]} &[{ ConvexRegionQ, BoundedRegionQ, RegionDimension, Module[{dim = RegionEmbeddingDimension[#]},  var = Symbol["x" <> ToString[#]] & /@ Range[dim];   dim] &, RegionMeasure, RegionCentroid, RegionProperty[#, var, "FastDescription"] &, RegionProperty[#, var, "ImplicitDescription"] &, RegionElement, LevelFunction[RegionProperty[#, var, "FastDescription"][[1, 2]]] & }] // Grid[Insert[#, {ConvexRegionQ, BoundedRegionQ, RegionDimension, RegionEmbeddingDimension, RegionMeasure, RegionCentroid, FastDescription, ImplicitDescription, RegionElement, LevelFunction}, 2]\[Transpose], Dividers -> All, FrameStyle -> GrayLevel[.8], Alignment -> Left] & // Quiet 

In addition of BoxRegion, other *Regions also seems to be used to declare regions:

Maybe some of them (LevelFunction) work only on "full-rank" regions?

Some regions look like special cases: