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?
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 $10^3.$