Colorize a Region

Question

My searches have lead nowhere on this one.
I have a MeshRegion (say, a 2D manifold embedded in 3D) and would like to plot a function on it: such that the color of a point at (x,y,z) is determined by a function of the coordinates.
I have found MeshCellStyle, which allows me to specify flat color per cell, but does give access to coordinates for a full ColorFunction-style. There's also RegionPlot3D, which might work (using (x,y,z) ∈ ℛ) for a full-dimensional submanifold but when passed a lower-dimensional object embedded in 3D the sampling algorithm never hits.

kglr · Accepted Answer

You can extract mesh coordinates and polygons of your MeshRegion and use them to construct a GraphicsComplex with VertexColors.
For example:
mreg = DiscretizeRegion[
       ImplicitRegion[-3 y - 4 x y^2 + 4 x z^2 + 4 x^2 z + 2 y^2 z^2 + z^3 == 0, 
         {x, y, z}],  MaxCellMeasure -> .001]

mc = MeshCoordinates[mreg];

polys = MeshCells[mreg, 2];

f[x_, y_, z_] := x y + z;

Graphics3D[{EdgeForm[], 
  GraphicsComplex[mc, polys, VertexColors -> (Hue[f@##] & @@@ Rescale[mc])]}, 
  Boxed -> False]

Use VertexColors -> (Opacity[.7, Hue[f@##]] & @@@ Rescale[mc]) to get

Colorize a Region

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