3 ms·

I don't think the writer is really doing anything much wrong. In the Wikipedia article on the topic "convex hull" is defined as you suggest for sets in abstract

I don't think the writer is really doing anything much wrong. In the Wikipedia article on the topic "convex hull" is defined as you suggest for sets in abstract, but for a finite set of points, as being the polygon (and not the interior of the polygon), which is consistent with the writer's terms and it is rather closer to both the intuitive understanding of the matter at hand and the algorithms under discussion. It seems to me that the term "convex hull" is ambiguous, even among geometers -- sometimes meaning the set and sometimes the boundary. A hull tends to be hollow, so I'd suggest that the boundary definition is nearer normal usage and intuition. In any event, the writer defines the terms and uses them as defined. I think the two algorithms he examines are interesting. The second is the one I first think of. The two methods are almost duals of each other. It seems to me that if you start by taking the top, left, right, and bottom points as your initial hull and then add on for any points not in that hull, it's almost certainly faster than either algorithm (especially in practice).

chaoxu 12y ago
Of course, using the most abstract definition is not the best idea, but anything equivalent to the standard terminology is ok. For example, in 2d, we can san say the convex hull of a finite set of points is the smallest enclosing convex polygon. The name convex hull is related to linear hull and conic hull, therefore defining it to not contain the interior is strange, but reasonable because in many applications we only care about the description of the convex hull by describing the polygon.
- Tloewald 12y ago
  Well in abstract there are plenty of possible spaces where a set can be bounded but its boundary does not exist within the set (e.g. consider the transcendentals between 0 and 1) so talking about the "hull" as a boundary is completely useless. Even so, picking the word "hull" was perhaps careless. A lot of people, even mathematicians, are confused on the definition of circle vs. sphere (the circle is the boundary, the sphere is the volume), so even if you use these terms "correctly" you probably want to define your understanding of them.

chaoxu 12y ago

Of course, using the most abstract definition is not the best idea, but anything equivalent to the standard terminology is ok. For example, in 2d, we can san say the convex hull of a finite set of points is the smallest enclosing convex polygon. The name convex hull is related to linear hull and conic hull, therefore defining it to not contain the interior is strange, but reasonable because in many applications we only care about the description of the convex hull by describing the polygon.

Tloewald 12y ago
Well in abstract there are plenty of possible spaces where a set can be bounded but its boundary does not exist within the set (e.g. consider the transcendentals between 0 and 1) so talking about the "hull" as a boundary is completely useless. Even so, picking the word "hull" was perhaps careless. A lot of people, even mathematicians, are confused on the definition of circle vs. sphere (the circle is the boundary, the sphere is the volume), so even if you use these terms "correctly" you probably want to define your understanding of them.