* We can `DiscretizeRegion` the `Cuboid[]` at first.

```
Clear[r];
r = TransformedRegion[DiscretizeRegion@Cuboid[], 
 Function[
 p, {p[[1]]*(-p[[1]] + p[[2]] + p[[3]]), 
 p[[2]]*(p[[1]] - p[[2]] + p[[3]]), 
 p[[3]]*(p[[1]] + p[[2]] - p[[3]])}]];
Region[r]
```
[![enter image description here][1]][1]

* Test another reg for example some polyhedron.

```
Clear[reg, r];
reg = PolyhedronData["Dodecahedron", "Region"];
r = TransformedRegion[DiscretizeRegion[reg], 
 Function[
 p, {p[[1]]*(-p[[1]] + p[[2]] + p[[3]]), 
 p[[2]]*(p[[1]] - p[[2]] + p[[3]]), 
 p[[3]]*(p[[1]] + p[[2]] - p[[3]])}]];
{reg, Region[r]}
```
[![enter image description here][2]][2]


 [1]: https://i.sstatic.net/kqz0R.png
 [2]: https://i.sstatic.net/2WDuc.png