Assigning a Color to each Point on the Plane

Convex hull mesh

coords = CoordinateBoundsArray[{{-10, 10}, {-10, 10}}];
in = Select[Flatten[coords, 1], Norm@# <= 5 &];
out = Select[Flatten[coords, 1], Norm@# > 5 &];

p1 = Cases[Normal@ConvexHullMesh[out]["Graphics"], _Polygon, Infinity];
p2 = Cases[Normal@ConvexHullMesh[in]["Graphics"], _Polygon, Infinity];

Graphics[{
  ColorData[97, 1], First@p1,
  ColorData[97, 2], First@p2
  }]

If anybody knows of a better way to turn a convex hull mesh into a polygon, please let me know.

Interpolation

Interpolation with interpolation order 0.

data = Join[{#, 1} & /@ in, {#, 2} & /@ out];
interp = Interpolation[data, InterpolationOrder -> 0];

DensityPlot[
 interp[x, y],
 {x, -10, 10},
 {y, -10, 10},
 PlotPoints -> 100
 ]

Using a higher interpolation order and rounding to get smoothed boundaries:

interp = Interpolation[data, InterpolationOrder -> 1];
DensityPlot[
 Round@interp[x, y],
 {x, -10, 10},
 {y, -10, 10},
 PlotPoints -> 100
 ]

VoronoiMesh method

points = Drop[data, None, -1];
mesh = VoronoiMesh[points];
polygons = MeshPrimitives[mesh, 2];
coloredpolygons = Map[{incircle @@ RegionCentroid[#], #} &, polygons];
Graphics[coloredpolygons]

VoronoiMesh automatically chooses the coordinates of the boundaries of the mesh, but you can also specify the boundaries via a second argument (VoronoiMesh[points, {{xmin, xmax}, {ymin, ymax}}]) if you want.

This method generalizes nicely to irregularly located points as well.

points = RandomReal[{-10, 10}, {200, 2}];
(* Remaining code as above *)

Note, however, that there is a bug in this code: the centroid of a Voronoi cell is not necessarily its "base point". This may make a difference in some cases, particularly if your base points are not uniformly distributed.

I will have to think if there's an easy way to color the Voronoi cells based on their base point, rather than their centroid. Complicating matters is that MeshPrimitives[VoronoiMesh[points]] appears to shuffle the order of the resulting polygons (i.e., the $i$th polygon in the result does not necessarily contain the $i$th element of points.)