# The two meanings of "convex hull"

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)?

• 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 $P$. 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. 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. 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) 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 :) Sep 28 '12 at 15:24

• 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. Sep 28 '12 at 15:33