I just want an idea, not a full analysis. Here is the object I'm talking about:
This object was given in my computational geometry class. If you don't understand the picture, this is how I would explain it:
- Take a bicycle's wheel, and lay it down flat on a table
- Take another wheel, and stand it up on the table, then lift the wheel a little bit (so there's a gap between the table and the wheel)
- But actually the 'standing' wheel is above the wheel lying flat on the table
Then, consider points on the perimeter of the wheel (but not the points in the center of the circle, I don't know why I drew them). We take $\frac{n}{2}$ points from both wheels.
Question: My professor said this object in 4d will have the 'complete graph' as the convex hull. I have no clue how to interpret this.
I know what convex hulls are. But suppose a set of points lies on a 2d plane. I don't see why the 2d convex hull is not a 3d convex hull as well. What I'm saying is that this object exists in 3d, so why can't it's 3d convex hull be the 4d convex hull as well?
I would also like to add, this was given as a quick example, and not really discussed in detail, so I might have mis-understood something (I have a suspicion that the 4d part is a mistake by me, and it's actually an example of a set of points in 3d whose convex hull is the complete graph). If I am mistaken, then the question is why is the convex hull the complete graph (as in all possible edges are in the convex hull)?
I don't even know the name of this object, so I can't google it.