Can anyone tell me the name of my shape? I'm trying to optimize something over the region in the positive octant of R^3 defined by:
(defn constraint [x,y,z]
  (or
   (and (<= x 10)
        (<= (+ y z) 10))
   (and (<= y 10)
        (<= (+ x z) 10))
   (and (<= z 10)
        (<= (+ x y) 10))))

i.e. the 10-ball of some sort of quasi-norm which is the max of the largest component and the sum of all the smaller ones.
I think it's one octant of a sort of octahedron with the faces triangulated and the centres moved towards the origin. 
It contains the cube (0,0,0) (5,5,5) and it seems to contain ~3/4 of the volume of the cube (0,0,0) (10,10,10).
I think the bashed-in octahedron has 24 faces, 14 vertices and 36 edges.
In 4d there's an analogous shape with volume ~36/64 of its bounding cube.
Does anyone know what the shape is called, and where I can find theorems about its volume? Also is there a name for the quasi-norm or a name for the class of things to which the quasi-norm belongs? And the family of shapes in n dimensions?
 A: These are just some thoughts.  It seems that we must have $0\leq x,y,z\leq10.$  Consider $x.$  If the first clause in the constraint is satisfied, then $x\leq10.$  If the second clause is satisfied, then $x+z\leq10,$ and since $x,z\geq0,\ x\leq10.$ Similarly if the third clause is satisfied.  So it seems that we can restate the constraints as $$0\leq x,y,z\leq10\\ \min\{x+y,x+z,y+z\}\leq 10$$
Each of the planes $x+y=0,\ x+z=10\ y+z=10$ cuts the cube in half, so the shape you are looking for is the union of three triangular prisms.  So I agree, it's a cube with a corner gouged out, but my spatial visualization isn't up to describing the shape of the gouge.  
The part gouged out is a convex shape satisfying $$
0\leq x,y,z\leq10\\
x+y>10\\
x+z>10\\
y+z>10$$ 
The constrained shape is not convex.  For example, the midpoint of $(10,7,3)$ and $(3,7,10)$ is $(6.5,7,6.5).$  I can't think of any non-convex polyhedra with well-known names except for the stellated polyhedra, so I think it's unlikely that this solid has a name.  I'm no expert though, so my opinion isn't worth much.
As to computing the volume of the shape, we could compute the volume of the gouged-out part with calculus and subtract from $10^3.$ 
A: I'm with @saulspatz: This shape does not have a name.

          


