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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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    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 :)2012-09-28

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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.2012-09-28