2
$\begingroup$

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)?

$\endgroup$
5
  • 5
    $\begingroup$ 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 $P$. $\endgroup$ Sep 28 '12 at 14:38
  • 1
    $\begingroup$ 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. $\endgroup$
    – hardmath
    Sep 28 '12 at 14:43
  • $\begingroup$ 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. $\endgroup$ Sep 28 '12 at 14:55
  • $\begingroup$ Same here - research background in computer graphics and computational geometry, only ever heard of it in terms of (2) $\endgroup$
    – fluffy
    Sep 28 '12 at 15:24
  • 1
    $\begingroup$ 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 :) $\endgroup$ Sep 28 '12 at 15:24
3
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$
    – GEdgar
    Sep 28 '12 at 15:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.