3D plot with Region seems inaccurate

Question

Region[ImplicitRegion[x/Max[y,z] == Max[y,z]/Abs[y-z] && x+y+z == 1,{x,y,z}], 
 PlotRange -> {{0.001, 0.999}, {0.001, 0.999}, {0.001, 0.999}}]

produces the following graph with gaps and incorrect linear segments:

Any advice would be very welcome.

Answer 1

Since the comment does not work for the new version 14.0 or 14.1, we do this similar with https://mathematica.stackexchange.com/a/307019/72111

plot = ContourPlot3D[x + y + z == 1, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}, 
  MeshFunctions -> 
   Function[{x, y, z}, x/Max[y, z] - Max[y, z]/Abs[y - z]], 
  Mesh -> {{0}}, MeshStyle -> Red, ContourStyle -> None, 
  BoundaryStyle -> None, MaxRecursion -> 6, PlotPoints -> 200]

DiscretizeGraphics[plot]

• Or rewrite the equation to x*Abs[y - z] == Max[y, z]*Max[y, z] and use Method -> "Semialgebraic" or Method->"MarchingCubes"

DiscretizeRegion[
   ImplicitRegion[
    Abs[y - z]*x == Max[y, z]*Max[y, z] && 
     x + y + z == 1, {{x, 0, 1}, {y, 0, 1}, {z, 0, 1}}], 
   MaxCellMeasure -> {"Length" -> 0.01}, 
   Method -> #] & /@ {"MarchingCubes", "Semialgebraic"}