I think the title of the section "In Euclidean geometry" is misleading since the content of the section is much more generel than this... —Preceding unsigned comment added by 130.226.45.151 ( talk) 17:48, 16 November 2009 (UTC)
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I took out the phrase "corresponding to regular polyhedra" because it didn't fit in the sentence and because it didn't convey important information. I guess one could rephrase the sentence to include the information that Platonic solids are regular polyhedra, but I didn't bother. --AxelBoldt
The following was added:
I do not understand this. Are we talking about efficiency frontiers in the sense of http://library.wolfram.com/webMathematica/MSP/Explore/Business/Frontier ? What does "maximized variable" refer to? AxelBoldt, Tuesday, June 11, 2002
In the most commonly discussed type of efficient frontier, one wishes to maximize average return and minimize variance of return. For this purpose the efficient frontier is the "northwest" hull of a plot of portfolios with mean return as the y coordinate and standard deviation of return as the x coordinate. The y-intercept, not surprisingly, is called the " risk-free rate".
I suppose the southeast hull could be called an "inefficient frontier". The same method of analysis is used in other optimization problems, using other numbers of variables. Some optimizations might seek to maximize all variables, or minimize some and maximize others. In any case, the efficient set is some convex portion of the outer hull of the points. This, of course, is assuming a large number of proposed solutions have been designed and had their specifications calculated and tabulated.
With regard to the discussion of convex sets, to see if I got this right, would it be correct to call a set of number whose coordinates in the complex plane form a convex polygon a convex set (or complex convex set)? Fredrik 15:30, 28 May 2004 (UTC)
I do not think topology has anything to do with convexity.... Tosha 12:19, 5 Jul 2004 (UTC)
Convex implies contractible, but after that, I probably agree. Charles Matthews 12:28, 5 Jul 2004 (UTC)
OK, the topology category isn't really useful. Charles Matthews 14:15, 6 Jul 2004 (UTC)
It may just be me, but shouldn't this page also include a simple picture to illustrate as well? The mathematical properties may not be what everyone is looking for when they come here.
This absolutely should be under Category:Topology too, but it needs to be changed to allow that: add the topological definition of convexity. This is what I came here looking for...if I cant find it here, where should I find it? Rob 00:11, 21 November 2005 (UTC)
Was thinking of adding the notion of star-convex sets. What does anyone think about including it here? It's not really a big enough topic to have its own page. cBuckley 12:30, 10 February 2006 (UTC)
EDIT: Added it anyway :-P cBuckley 13:40, 10 February 2006 (UTC)
Convex polygon would fit in here greatly. -- Abdull 14:23, 17 May 2006 (UTC)
I would say that we should mention quasi convex in the article too. & strongly, strictly quasi convex. & semi convex. Jackzhp ( talk) 17:08, 13 September 2008 (UTC)
It is erroneous to have a picture with the legend/title "non-convex (i.e., concave)". A halfspace is a convex set whose complement is convex, for example.
It might be useful to discuss quasi-convex sets and pseudo-convex sets, from the standpoint of (Rockafellarian) variational analysis, by discussing the epigraphs and lower levelsets of the ( convex-analytic) indicator functions (or set-theoretic indicator functions) for such sets, i.e., the functions's quasiconvexity or pseudoconvexity. (This suggestion accords with the comments of some previous discussants.) However, such definitions are incompatible with the usage of several complex variables. Kiefer.Wolfowitz ( talk) 18:39, 24 June 2009 (UTC)
Yes, the discussion in the original "concave set" page proved quite controversial. I am of the opinion that the term "concave set" is erroneous to begin with, and I believe the clear majority agree with this. I've merged the corrected text for "concave set" into this one. I created a "non-convex set" subsection so that the redirection can point to that specific discussion. Mcgrant ( talk) 13:53, 25 October 2014 (UTC)
While the article Concave set is very comprehensive as it currently stands, maybe it would be better deleted and integrated into this article. I don't know how to do any of this, though
I've done this merging. Mcgrant ( talk) 13:51, 25 October 2014 (UTC)
Convex curve is logically contained within convex set topic. Also, wiki entry on the former is literally just one statement. IT seems appropriate to merge it with convex set. Mittgaurav ( talk) 08:40, 31 December 2011 (UTC)
Perhaps this picture of the german wikipedia is useful in the article: http://de.wikipedia.org/wiki/Datei:KonkaveFiguren.png -- Flegmon ( talk) 11:58, 26 April 2012 (UTC)
"Complex convexity" is just a special case of "convex set" where you're operating in C^n - no need to have a whole other article. Enterprisey ( talk!) 04:44, 8 September 2018 (UTC)
This is gibberish. For starters, subset is singular, so the verb should be "intersects" - but then we are left with "intersects into a line segment" which is not any kind of maths jargon I have ever heard. 2A01:CB0C:CD:D800:C8A2:EE7F:D271:DFC6 ( talk) 08:04, 4 June 2020 (UTC)
"The Minkowski sum of two compact convex sets is compact." -- compact sets are not introduced in this article. Cuppoo ( talk) 02:42, 13 December 2023 (UTC)
I think the title of the section "In Euclidean geometry" is misleading since the content of the section is much more generel than this... —Preceding unsigned comment added by 130.226.45.151 ( talk) 17:48, 16 November 2009 (UTC)
This
level-5 vital article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
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I took out the phrase "corresponding to regular polyhedra" because it didn't fit in the sentence and because it didn't convey important information. I guess one could rephrase the sentence to include the information that Platonic solids are regular polyhedra, but I didn't bother. --AxelBoldt
The following was added:
I do not understand this. Are we talking about efficiency frontiers in the sense of http://library.wolfram.com/webMathematica/MSP/Explore/Business/Frontier ? What does "maximized variable" refer to? AxelBoldt, Tuesday, June 11, 2002
In the most commonly discussed type of efficient frontier, one wishes to maximize average return and minimize variance of return. For this purpose the efficient frontier is the "northwest" hull of a plot of portfolios with mean return as the y coordinate and standard deviation of return as the x coordinate. The y-intercept, not surprisingly, is called the " risk-free rate".
I suppose the southeast hull could be called an "inefficient frontier". The same method of analysis is used in other optimization problems, using other numbers of variables. Some optimizations might seek to maximize all variables, or minimize some and maximize others. In any case, the efficient set is some convex portion of the outer hull of the points. This, of course, is assuming a large number of proposed solutions have been designed and had their specifications calculated and tabulated.
With regard to the discussion of convex sets, to see if I got this right, would it be correct to call a set of number whose coordinates in the complex plane form a convex polygon a convex set (or complex convex set)? Fredrik 15:30, 28 May 2004 (UTC)
I do not think topology has anything to do with convexity.... Tosha 12:19, 5 Jul 2004 (UTC)
Convex implies contractible, but after that, I probably agree. Charles Matthews 12:28, 5 Jul 2004 (UTC)
OK, the topology category isn't really useful. Charles Matthews 14:15, 6 Jul 2004 (UTC)
It may just be me, but shouldn't this page also include a simple picture to illustrate as well? The mathematical properties may not be what everyone is looking for when they come here.
This absolutely should be under Category:Topology too, but it needs to be changed to allow that: add the topological definition of convexity. This is what I came here looking for...if I cant find it here, where should I find it? Rob 00:11, 21 November 2005 (UTC)
Was thinking of adding the notion of star-convex sets. What does anyone think about including it here? It's not really a big enough topic to have its own page. cBuckley 12:30, 10 February 2006 (UTC)
EDIT: Added it anyway :-P cBuckley 13:40, 10 February 2006 (UTC)
Convex polygon would fit in here greatly. -- Abdull 14:23, 17 May 2006 (UTC)
I would say that we should mention quasi convex in the article too. & strongly, strictly quasi convex. & semi convex. Jackzhp ( talk) 17:08, 13 September 2008 (UTC)
It is erroneous to have a picture with the legend/title "non-convex (i.e., concave)". A halfspace is a convex set whose complement is convex, for example.
It might be useful to discuss quasi-convex sets and pseudo-convex sets, from the standpoint of (Rockafellarian) variational analysis, by discussing the epigraphs and lower levelsets of the ( convex-analytic) indicator functions (or set-theoretic indicator functions) for such sets, i.e., the functions's quasiconvexity or pseudoconvexity. (This suggestion accords with the comments of some previous discussants.) However, such definitions are incompatible with the usage of several complex variables. Kiefer.Wolfowitz ( talk) 18:39, 24 June 2009 (UTC)
Yes, the discussion in the original "concave set" page proved quite controversial. I am of the opinion that the term "concave set" is erroneous to begin with, and I believe the clear majority agree with this. I've merged the corrected text for "concave set" into this one. I created a "non-convex set" subsection so that the redirection can point to that specific discussion. Mcgrant ( talk) 13:53, 25 October 2014 (UTC)
While the article Concave set is very comprehensive as it currently stands, maybe it would be better deleted and integrated into this article. I don't know how to do any of this, though
I've done this merging. Mcgrant ( talk) 13:51, 25 October 2014 (UTC)
Convex curve is logically contained within convex set topic. Also, wiki entry on the former is literally just one statement. IT seems appropriate to merge it with convex set. Mittgaurav ( talk) 08:40, 31 December 2011 (UTC)
Perhaps this picture of the german wikipedia is useful in the article: http://de.wikipedia.org/wiki/Datei:KonkaveFiguren.png -- Flegmon ( talk) 11:58, 26 April 2012 (UTC)
"Complex convexity" is just a special case of "convex set" where you're operating in C^n - no need to have a whole other article. Enterprisey ( talk!) 04:44, 8 September 2018 (UTC)
This is gibberish. For starters, subset is singular, so the verb should be "intersects" - but then we are left with "intersects into a line segment" which is not any kind of maths jargon I have ever heard. 2A01:CB0C:CD:D800:C8A2:EE7F:D271:DFC6 ( talk) 08:04, 4 June 2020 (UTC)
"The Minkowski sum of two compact convex sets is compact." -- compact sets are not introduced in this article. Cuppoo ( talk) 02:42, 13 December 2023 (UTC)