# Convert ConvexHull to Inequality

I have the following convex hull.

P0={0,0,0};
P1={1,0,0};
P2={0,1,0};
P3={0,0,1};

ConvexHullMesh[{P0,P1,P2,P3}]


Now I want the region algebraically as equality of the convex hull generated, in this case x+y+z<1&&x>0&&y>0&&z>0. Is there any automatic way to get it from Mathematica by converting the convex hull generated as a region and then convert it into inequality?

chm = ConvexHullMesh[{P0, P1, P2, P3}];

ClearAll[regFunc]

regFunc[{x, y, z}] := FullSimplify @ RegionMember[Rationalize @
MeshPrimitives[DiscretizeRegion[chm, MaxCellMeasure -> ∞], 3][[1]]] @ {x, y, z}

regFunc @ {x, y, z}

 x + y + z <= 1 && z >= 0 && y >= 0 && x >= 0


Also

dm = DelaunayMesh[{P0, P1, P2, P3}];

ClearAll[regFunc2]

regFunc2[{x_, y_, z_}] := FullSimplify @
RegionMember[Rationalize @ First @ MeshPrimitives[dm, 3]] @ {x, y, z};

regFunc2 @ {x, y, z}

 x + y + z <= 1 && z >= 0 && y >= 0 && x >= 0

• Thanks a lot. Just another question: Can this be done for a 4-d convex hull? Of course, Mathematica cannot evaluate the convex hull for a 4-d case. But is there any way to get the inequality representation of the convex hull? The type of convex hull I work with has 5 points for a 4-d case and so is very simple. Say for example {0,0,0,0},{1,0,0,0}, {0,1,0,0},{0,0,1,0},{0,0,0,1}. Is there any way to get it just like the 3d case you showed in Mathematica itself? Jun 14, 2020 at 4:32
• @SumitBanik, thank you for the accept. Don't know how the nD case can be handled.
– kglr
Jun 14, 2020 at 4:47
• Alright.Thanks a lot Jun 14, 2020 at 4:55
• An nD example can be found here. Jun 14, 2020 at 11:36
• When I several convex hulls defined in an array R[i], where say Jun 15, 2020 at 2:38