Polygon vertices
In[]:=
pts={1,1},{-1,1},{-1,0},
-1
2
,0,0,
-1
2
,{0,-1},{1,-1};
Polygon with the above vertices
In[]:=
poly=Polygon[pts]
Out[]=
Polygon
Number of points: 7
Embedding dimension: 2

Reduce the Logical Not of a Point Which Is An Element of the Polygon
In[]:=
outerCons=Reduce[!Element[{x,y},poly],{x,y},Reals]
Out[]=
x<-1||-1≤x≤-
1
2
&&(y<0||y>1)||-
1
2
<x<0&&y<
1
2
(-1-2x)||y>1||(0≤x≤1&&(y<-1||y>1))||x>1
Region Plot of the Reduce Element Constraints
In[]:=
rp=RegionPlot[outerCons,{x,-2,2},{y,-2,2},ImageSize->Small]
Out[]=
Perform a Logical Expand on the Constraints and Convert the Head from Or to List for Plotting
In[]:=
lgExOuterCons=List@@LogicalExpand@outerCons
Out[]=
y>1&&-
1
2
<x&&x<0,y>1&&-1≤x&&x≤-
1
2
,y>1&&0≤x&&x≤1,-
1
2
<x&&x<0&&y<
1
2
(-1-2x),y<-1&&0≤x&&x≤1,y<0&&-1≤x&&x≤-
1
2
,x>1,x<-1
In[]:=
Show[Table[RegionPlot[lgExOuterCons[[i]],{x,-2,2},{y,-2,2}],{i,Length[lgExOuterCons]}],ImageSize->Small]
Out[]=