How do I plot 3D intersections of a system of inequalities using Matlab or Mathematica?

Question

My question is inspired by the problem described here. Let us consider for example the following system of inequalities ($x,y,z\in\mathbb{R}$):

• $x+y+z>0$
• $x^3+y^3+z^3<0$
• $x^5+y^5+z^5>0$

How can I plot the intesections of these areas (my hope is to be able to visualize that in a region around the origin $(0,0,0)$ not all conditions are met.

What I tried so far is illustrating the regions in 3D using the following code:

[X,Y,Z]=meshgrid(-0.1:0.01:0.1,-0.1:0.01:0.1,-0.1:0.01:0.1);
ineq1 = (X+Y+Z > 0);
ineq2 = (X.^3+Y.^3+Z.^3 < 0);
ineq3 = (X.^5+Y.^5+Z.^5 > 0);
colors = zeros(size(X))+ineq1+ineq2+ineq3;
scatter3(X(:),Y(:),Z(:),3,colors(:),'filled')

As a result at least I get a rough idea of the situation:

It would be nice to see a contour (line/area) where that conditions coincide.

Answer 1

Try

RegionPlot3D[x + y + z > 0 && x^3 + y^3 + z^3 > 0 && x^5 + y^5 + z^5 > 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, PlotPoints -> 40]