Talk:Convex hull

message: A change today added a mention of convex hulls in infinite dimensional vector spaces. But, if one has a convex hull of infinitely many points in an infinite dimensional space, does one take all convex combinations for which the set of weights forms a series summing to one, or only those convex combinations with finite support? That is, is (1/2,1/4,1/8,...) a member of the convex hull of (1,0,0,...), (0,1,0,...), (0,0,1,...), ...? The definition of a convex hull as the minimal convex set containing a set of points, and the definition of a convex set as containing the line segment connecting any two of its points, leads to the finite support version. But, on the other hand, the definition of convex combination and of a convex hull as the set of all convex combinations seems to be stated in a way that would imply the other version. Perhaps this should be stated here more explicitly? —David Eppstein (talk) 00:12, 24 May 2008 (UTC)
replies: text: I see no bigger problem with a vector space of infinite dimension than with a vector space of finite dimension. In each case there is an infinite number of points, and the convex hull is the set of sums of convex combinations of finite number of points. The number of points in a convex combination is not tied in any way to the dimension of the space. Oleg Alexandrov (talk) 04:53, 24 May 2008 (UTC)
replies: text: I agree that's the right definition. But our convex combination article doesn't clearly state that it's a combination of only finitely many; one could read it as allowing infinitely-supported combinations. —David Eppstein (talk) 06:45, 24 May 2008 (UTC)
replies: text:  Well, there is the number n there, which implies that the number of points is finite. I clarified that more there. Also, the possibility of an infinite amount of points is ruled out by the fact that it is impossible to talk about an infinite sum unless you have some kind of norm/topology which is not always the case in real vector spaces. Oleg Alexandrov (talk) 14:55, 24 May 2008 (UTC)
replies: text: Much better. I still have a quibble: now convex combinations are defined only for finite sets of points, while here we say that the convex hull is the set of convex combinations of all of the points, so we can only define convex hulls of finitely many points. Two possible fixes: (1) in the convex combination article, define a convex combination of infinitely many points to be a weighted sum with finite support, or (2) here, define the convex hull as the set of convex combinations of finite subsets of the points. I think (1) would be preferable, but I'm not sure about it. —David Eppstein (talk) 15:34, 24 May 2008 (UTC)
replies: text:  David, I reverted your edit since I believe a definition using finite support convex combinations of an infinite number of points is not that usual (I think). I understand your point after I read this convex hull article. The recent changes to it made the definition unclear. The convex hull is not the convex combination of all points in the space, but rather the set of all possible convex combinations of finite tuples of points. I hope after my revert to the text this is more clear. Oleg Alexandrov (talk) 04:05, 25 May 2008 (UTC)
replies: text:  Convexity is a special case of sequential convexity, which is a special case of measure convexity; all of these inclusions are strict (and important). For sequential convexity, see Topology and Normed Spaces by G[raham]. J[ames]. O[scar] Jameson for sequential convexity. For measure convexity (c.f., Choquet theory), see  Choquet order and simplices: with applications in probabilistic models by Gerhard Winkler (Springer Verlag, Lecture notes in mathematics ; 1145). Kiefer.Wolfowitz (talk) 17:32, 26 November 2009 (UTC)
thread_title:  Infinite dimensional? 

message: Forgive me (and correct me) if i'm not following appropriate protocol, however, i've noticed that the text in the "Applications" section, starting with the words "For example, consider the problem" and continuing through to "from one hull vertex to another", is word-for-word identical to a passage in the book, The Algorithm Design Manual by Steven Skiena (1998, Springer-Verlag New York), page 351 of the hardcover edition (section 8.6.2, "Convex Hull").
This text first appeared (with an additional sentence of quoted text from the book) in this revision, dating back to mid-2006. I've removed the content for the moment as a quick fix. Please advise if this is the incorrect course of action. Jmelesky (talk) 23:42, 22 June 2008 (UTC)
replies: text: Thanks for spotting this! And it wasn't exactly a perfect formulation anyway, since the careless first sentence suggested that the diameter is a pair of points, not their distance. For lack of context it wasn't particularly clear either, and it may be best to drop this particular aspect altogether. --Hans Adler (talk) 00:13, 23 June 2008 (UTC)
replies: text: I think it's an important aspect of the problem, and have added back a sentence that states it again without being a copyvio. Thanks for catching this. —David Eppstein (talk) 00:31, 23 June 2008 (UTC)
thread_title:  Copyright violation? 

message: Why do editors keep removing the reference to Carathéodory's theorem, without offering reasons or seeking consensus (at least since I've started watching this page)?
Of course, Carathéodory's theorem (Farkas, Steinitz, etc.)  exemplifies Stigler's "law of eponymy".
Kiefer.Wolfowitz (talk) 14:57, 31 January 2010 (UTC)
thread_title: [[Carathéodory's theorem (convex hull)|Carathéodory's theorem]]

message: There is a mention of Kallay's incremental algorithm, with no citation.  I've looked for this algorithm for over an hour, and I can't find it.  Is there a mistake or do I suck at finding things?  — Preceding unsigned comment added by 67.80.148.73 (talk) 10:27, 13 March 2012 (UTC)
replies: text: Try Gil Kalai, and fix the typo if my guess is correct, please!   Kiefer.Wolfowitz 11:01, 13 March 2012 (UTC)
thread_title:  Kallay? 

message: I miss the definition of the lower and upper convex hull. The term is used for example in David Gross: Basic Structures of Function Field Arithmetic p. 39 def. 2.7. A definition I found in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.69.1152&rep=rep1&type=pdf 
But I know no texbook as a source. I would be very thankfull if somebody could add this. --Flegmon (talk) 11:36, 16 April 2012 (UTC)
replies: text: Oh and it ist used in Newton_polygon --Flegmon (talk) 12:41, 16 April 2012 (UTC)
replies: text: Do you know of an encyclopedia article that discusses such "convex hulls"? Are they hull operators, convexity systems, antimatroids?  Kiefer.Wolfowitz 16:21, 16 April 2012 (UTC)
replies: text: I don't know how you want to abstract them, but they're very standard in computational geometry. I'll see if I can find a source in the computational geometry texts I have. —David Eppstein (talk) 16:48, 16 April 2012 (UTC)
thread_title:  lower, upper convex hull 

message: The definition of complex hull given in the first sentence of the article implies (at least for those who understand the definition of convex set) that a convex hull is not a boundary, but a solid object.
Since "hulls" and "envelopes" are kinds of hollow containers in plain English, it is quite difficult to accept that in mathematics a convex "hull" or "envelope" is a solid (non-hollow) object, rather than an external surface or boundary. This is made even more difficult by the rubber-band example given in the introduction ("the convex hull may be visualized as the shape formed by a rubber band stretched around X"), which is not compatible with the above mentioned definition. According to that definition, the covex hull would be the space bounded by the rubber band, rather than "the shape formed by" it.
Are you sure that the reliable literature consistently defines a convex hull as a convex set (rather than its boundary)? If this is true, then the introduction should warn readers that in mathematics a hull is not a container... Otherwise, the introduction should say that according to some (reliable) authors the hull is just a boundary (similar to an elastic band), and according to some others it is a convex set.
In this context, it may be also useful for editors to notice that in English, whatever is inside a container is simply "contained in" it. However, the expressions "contained in" or "included in" are used in set theory to mean "member of", or "part of" the set. Therefore, if a container (such as a bottle or the hull of an airplane) is modeled as a set of particles, whatever is inside the container (a liquid or the passengers of an airplane) is not "contained in" it, as it is not a part of it.
Paolo.dL (talk) 15:04, 7 January 2013 (UTC)
replies: text: The word "hull" in mathematics can be used with very abstract meanings — see e.g. injective hull. —David Eppstein (talk) 15:37, 7 January 2013 (UTC)
thread_title:  "Hull" as a container 

message: * The union of all simplices with vertices in X.
So there can't be a pathological point set such that the union of simplices has a hole? —Tamfang (talk) 03:31, 1 March 2013 (UTC)
replies: text: Keep reading.  In the paragraph that follows you'll find a link to Carathéodory's_theorem_(convex_hull).  --JBL (talk) 04:08, 1 March 2013 (UTC)
replies: text: Which seems obvious enough that I now wonder what prompted my question. —Tamfang (talk) 07:28, 10 February 2024 (UTC)
thread_title:  simplices 

message: The sentence "In fact, according to Carathéodory's theorem, if X is a subset of an N-dimensional vector space, convex combinations of at most N + 1 points are sufficient in the definition above" is misleading, as it appears to suggest that there exists set of $n+1$ vectors, such that convex combinations of them give the convex hull of the original set. Yet, this only gives a simplex and it is the union of many of these that give the complex hull of the original set, which implies that we need more than $n+1$ vectors. Rereading the sentence, perhaps misleading is too generous? 130.95.80.94 (talk) 02:49, 24 October 2014 (UTC)
thread_title:  Definition ambiguity 

message: Some of these sound very interesting, but they are unsupported.  Because this section is about applications, rather than finding citations that quote an author saying something like "convex hulls are useful for <blah>", it might be more helpful and interesting to the reader to link to, say, a paper that actually uses a convex hull for the listed application.  Just an idea to consider.  If I remember, I'll help add citations later. --166.20.224.10 (talk) 19:49, 24 November 2014 (UTC)
replies: text: I've added a specific application for home range with citation. —Mrwojo (talk) 23:07, 7 March 2015 (UTC)
thread_title:  No citations in the Applications section 

message: I don't see how it's OR to include the use in materials science when this is cited to book published by Cambridge University Press and Springer. A quick search would show that this is a well established topic in science , hence I'm wondering if you actually read the sources or instinctively lumped my edits in with the recent edits by another user. Pieceofmetalwork (talk) 08:48, 22 June 2022 (UTC)
thread_title:  In materials science 

message: This article claims that the convex hull of a compact set is compact, but this appears to be false without assuming some extra conditions: https://math.stackexchange.com/questions/2017113/is-the-convex-hull-of-a-compact-set-compact 196.3.50.248 (talk) 10:16, 22 December 2022 (UTC)
replies: text: This article is primarily about convex hulls in finite dimensional Euclidean spaces. In more general kinds of spaces like the one of that example, more general kinds of things can happen. —David Eppstein (talk) 07:50, 23 December 2022 (UTC)
thread_title:  Convex hull of a compact set