>>A concrete example would be the sum of the tetrahedron T with vertices
>><0,0,0>, <1,0,0>, <0,1,0> and <0,0,1> with "its negative" -T with 
>>vertices <0,0,0>, <-1,0,0>, <0,-1,0> and <0,0,-1>.
>>The outcome is a polyhedron (polytope) with vertices
>><-1, 0, 0>, <0, -1, 0>, <0, 0, -1>, <1, 0, 0>, <1, -1, 0>, <1, 0, -1>,
>><0, 1, 0>, <-1, 1, 0>, <0, 1, -1>, <0, 0, 1>, <-1, 0, 1>, <0, -1, 1>.
> 
> ...
> 
> Why isn't <0, 0, 0> in the outcome list ?
> 

Thinking of a polytopes as the "convex hulls" 
(http://mathworld.wolfram.com/ConvexHull.html) of their vertices, the 
sum of two polytopes A and B is the convex hull of all points a+b, where 
a is a vertex of A and b is a vertex of B.

In the example above <0, 0, 0> is a point a+b (of 16 possible 
combinations) in the interior of the convex hull. So it is not one of 
the 12 vertices of this polytope.

I hope this explains it.

Achill

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