# Talk:Convex set

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WikiProject Mathematics (Rated C-class, High-priority)

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)

## ?

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

One application of convex hulls is found in efficiency frontier analysis. Efficiency is assumed to be a monotonic function of each of finitely many of real variables. Each one of finitely many data points is in exactly one hull, and is considered more efficient than all data points in hulls contained within its own hull. A particle whose velocity vector has a value of a for all coordinates representing maximized variables, and a value less than a for all minimized variables, will pass through the hulls in increasing order of efficiency.

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)

## Category:Topology

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)

I only wanted to say that it should not be in Category:Topology (hope you agree) Tosha 11:13, 6 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)

## Star Convexity

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

Convex polygon would fit in here greatly. --Abdull 14:23, 17 May 2006 (UTC)

## convex vs. quasi convex

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)

As far as I know, quasiconvexity is a concept that generalizes convex functions, not convex sets. The convex function article does mention quasiconvex functions in its see-also section, but I suppose that could be better integrated into the text of the article. —David Eppstein (talk) 21:09, 28 November 2008 (UTC)

## "Concave set" versus a "set with a concavity" versus "a nonconvex set"

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)

## Concave Sets

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)

The only two 'authorities' you cite, Mcgrant, are from econometrics authors. While I have some bias, I'm generally skeptical of any mathematics done by 'economists', as I tend to think the vast majority wouldn't disclose their biases. Most math written down by 'economists' is intended to obfuscate the fact that they have no clue what they are talking about from people with math anxiety. To that end, and given my newly assigned interest in concave polyhedra, and more generally polytopes and sets, I am leaving notice that I may soon change the redirect and cite actual "authorities" in mathematics, rather than econometrics hacks. -66.169.240.117 (talk) 16:45, 3 January 2016 (UTC)
I am more than happy to see more references here. I am not an economist myself. Mcgrant

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)

It is not clear from that one statement if the boundary of a convex set in any number of dimensions is called a convex curve, or if what is meant is rather a convex function. I think a reference is needed to sort this out. Isheden (talk) 11:06, 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)

I think the existing picture that illustrates a non-convex set is better than these examples. It is misleading to indicate the midpoints of the connecting lines, since it is not enough that the midpoint is in the set. Rather, all points on the connecting line have to be in the set. In the present example, all points that lie outside the set or highlighted in red. Isheden (talk) 12:15, 26 April 2012 (UTC)

## Proposed merge with Complex convexity

"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)

I think C^n is different enough from R^n, and (real) convexity is basic enough, that this would be overly confusing and WP:TECHNICAL as an addition to the convex set article. And I don't see why we should privilege this notion over the five other generalizations of convexity already listed (in summary style) in the "Generalizations and extensions for convexity" section of the convex set article; what makes it deserve full elaboration here, rather than another summary-style entry, besides the fact that the existing complex convexity article is so stubby and incomplete? —David Eppstein (talk) 05:51, 8 September 2018 (UTC)
David Eppstein, I'm proposing that Complex convexity be merged into this article. I strongly agree with you that it's weird to have special treatment for this topic - I would rather complex convexity be listed in that section, with Complex convexity being a redirect to it. Enterprisey (talk!) 06:35, 8 September 2018 (UTC)
To be treated equally, it should be handled in summary style, with a pointer here to a separate and more complete article at the existing title. Anything else is special treatment. —David Eppstein (talk) 06:37, 8 September 2018 (UTC)
I see what you're saying. I'm not sure that the sources are there to support an article rather than a redirect, but I don't feel particularly strongly about this. I've put a link in the "See also" section for the time being. Enterprisey (talk!) 06:44, 8 September 2018 (UTC)
The reply may be too late, but I think it's related to Pseudoconvexity. In that article, it was linked to plurisubharmonic functions. --SilverMatsu (talk) 05:49, 11 January 2022 (UTC)
I don't have the math knowledge to know how that topic relates to this one, so feel free to do whatever. Enterprisey (talk!) 02:09, 16 January 2022 (UTC)

## is a subset that intersect every line into a single line segment (possibly empty).

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)