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RegionMember[RegionBoundary[Cuboid[]]]

Works, but 

RegionMember[RegionBoundary[RegionDifference[Cuboid[], Ball[]]]]

Returns nothing, even though the region RegionBoundary[RegionDifference[Cuboid[], Ball[]]] is a Region[] 

RegionQ[RegionBoundary[RegionDifference[Cuboid[], Ball[]]]]
(* True *)

And it even RegionBoundary[RegionDifference[Cuboid[], Ball[]]] looks like a region!

Region[RegionBoundary[RegionDifference[Cuboid[], Ball[]]]]

enter image description here

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  • $\begingroup$ Region[RegionBoundary[RegionDifference[Cuboid[], Ball[]]]] works for me. Is your syntax RegionMember[RegionUnion[RegionBoundary[RegionDifference[Cuboid[], Ball[]]]]] correct? $\endgroup$
    – user64494
    Commented May 19, 2020 at 16:05
  • $\begingroup$ Try finding the RegionMemberFunction[] of it by using RegionMember[] - that seems to be issue...I noticed the title was misleading - so I have edited it. $\endgroup$
    – Tomi
    Commented May 19, 2020 at 16:06
  • $\begingroup$ Seems like a bug to me. RegionMember should not be returning {}. $\endgroup$
    – Greg Hurst
    Commented May 19, 2020 at 16:18
  • $\begingroup$ I mean RegionUnion[RegionBoundary[RegionDifference[Cuboid[], Ball[]]]]. Hope I am clear. $\endgroup$
    – user64494
    Commented May 19, 2020 at 16:23
  • $\begingroup$ (And btw you don't need to use RegionUnion on one argument.) $\endgroup$
    – Greg Hurst
    Commented May 19, 2020 at 16:23

2 Answers 2

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We can find a workaround for semialgebraic regions through CylindricalDecomposition.

regionBoundary[reg_?RegionQ] := 
  Module[{x,y,z},
    ImplicitRegion[
      CylindricalDecomposition[RegionMember[reg, {x, y, z}], {x, y, z}, "Boundary"], 
      {x, y, z}
    ]
  ]

RegionMember[regionBoundary[RegionDifference[Cuboid[], Ball[]]]] // Head

RegionMemberFunction
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Not really a solution, but a work around is to wrap the shapes in BoundaryDiscretizeRegion[]: 

RegionMember[RegionDifference[BoundaryDiscretizeRegion[Cuboid[]], BoundaryDiscretizeRegion[Ball[]]]]

Returns a RegionMemberFunction

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  • $\begingroup$ Also RegionMember[ RegionDifference[BoundaryDiscretizeRegion[Cuboid[]], BoundaryDiscretizeRegion[Ball[]]], {1, 2, 3.1}] performs False. $\endgroup$
    – user64494
    Commented May 19, 2020 at 17:01

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