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Given a finite set of points $P$, we can think of two meanings for the term "convex hull of $P$:"

  1. The set of convex combinations of points from $P$, $\sum \alpha_i p_i$ s.t. $\sum \alpha_i = 1 \wedge \forall \alpha_i \geq 0$.
  2. The points $P' \subseteq P$ which are "active" in the convex hull; that is, the removal of which changes the set of points from (1).

Question: Is there a standard way to refer specifically to (1) or (2)?

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I have never heard of convex hull as in 2, only as in 1. The points in 2 I would call the extreme points of the convex hull of $P$. – Harald Hanche-Olsen Sep 28 '12 at 14:38
This confusion in terminology is also a peeve of mine. The word "hull" means an exterior, and there's a perfectly good phrase "convex closure" for (1). Nonetheless the use of convex hull to mean (1) predominates except when discussion turns to algorithms for finding the extreme points (2) as a "representation" of the convex hull. – hardmath Sep 28 '12 at 14:43
Indeed, I come from CS, where (2) dominates "convex hull". Funny it's unheard-of in Math :) @hardmath turn your comment into an answer to be accepted. "convex closure" is what I was looking for, which appears to be a known synonym for CH. – user1071136 Sep 28 '12 at 14:55
Same here - research background in computer graphics and computational geometry, only ever heard of it in terms of (2) – fluffy Sep 28 '12 at 15:24
Intrigueing. With interpretation (2) you seem to require the convex hull to be a hull in some everyday sense, but you don't require it to be convex any more :) – Hagen von Eitzen Sep 28 '12 at 15:24
up vote 3 down vote accepted

I prefer "convex closure" for the collection of all convex combinations (1). The phrase convex hull is slightly more concise, but the etymology of "hull" carries a meaning of the outer covering (as in the hull of a boat or husk of a seed), so it seems a pity to me not to reserve this for what mathematicians instead like to call the "extremal boundary points". It's an important enough algorithmic (computational geometry) topic to deserve a shorter name.

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If there is a topology involved, too, mathematicians may speak of the "convex hull" and the "closed convex hull". Of course they use these terms even if $P$ is infinite. If they saw "convex closure" they may not know which is would mean. – GEdgar Sep 28 '12 at 15:33

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