# How to plot this set $M\subset\mathbb{R}^3$?

Is there a way to plot the following set with Mathematica?

$$M=\left\{ \left(x,y,z\right)\mid x\in\mathbb{R}\:,\:y=\left|x\right|\:,\:0\leq y

I think it's suppose to look something like this:

$$y=|x|$$ is plotted just for reference, and also $$(0,0,0)\notin M$$ (the origin is not part of the region), but I don't mind if the plot includes this point.

We define regionplot[] to take two arguments: w is the region, and p is the number of points passed to PlotPoints.

regionplot[w_, p_] := RegionPlot3D[w, {x, -1, 1}, {y, -1, 1}, {z, -1, 1},
BoundaryStyle -> Directive[LightBlue, Thick], Mesh -> None,
PlotPoints -> p, PlotRange -> All, AxesLabel -> {x, y, z},
PerformanceGoal -> "Quality", MaxRecursion -> 5,
PlotRange -> {{-1, 1}, {0, 1}, {0, 1}}];

We found that plotting the pair of walls with no thickness, leaves quite a gap between the walls. Giving the walls a little thickness, and using p=200 for a nice pinstripe pattern:

regionplot[0 < y < z && Abs[x] - .001 < y < Abs[x] + .001, 200]

• is there a way to smooth the surface and to get rid of the gap there?
– Jon
Mar 19, 2019 at 11:59
• Yes, I know, it is disappointing! Using PlotPoints-> 1000 instead of 100 gives a nice, light blue wall. Unfortunately, it only plots one wall! (Mathematica 11.2.0.0) A workaround could be to plot to walls individual (wall1 and wall2 and then use Show[]: as in Show[wall1,wall2].
– mjw
Mar 19, 2019 at 12:49
• Comment correction: "to walls" ---> "the two walls"
– mjw
Mar 19, 2019 at 13:41
• I wasn't speaking about the gap between the lines forming the walls, but the gap between the two walls. This gap
– Jon
Mar 19, 2019 at 13:52
• Right! I also was speaking about the gap between the walls when I wrote there is still a bit of a gap. I don't mind the pinstripes, and with more PlotPoints, I think they disappear.
– mjw
Mar 19, 2019 at 14:32

This can be done as follows.

Region[ImplicitRegion[y == RealAbs[x] && 0 <= y < z, {x, y, z}]]