Talk:Convex function

message: Should the definition read "for all x,y" instead of for any?
:To anyone used to reading math (and maybe most who aren't), it's the same thing. But the change might improve clarity some small amount. Dchudz 17:08, 15 November 2006 (UTC)
replies: text: You are mistaken about 'for any'. 'For any' is equivalent to ∀, 'for some' is ∃.--92.219.6.7 (talk) 09:02, 30 July 2017 (UTC)
thread_title: Correct Definition

message: Honestly, think that's a bad idea. Although the two concepts are so completely intertwined as to be two sides of the same coin, people come to Wikipedia for answers, not answer-hunting (reference the post above). I suppose it's possible that the merge could be done in such a way as to not be too confusing, but take a look at supermodular to see what can happen when things get thrown together (read the title, and then the definition; but then I suppose I'm supposed to fix it, not just sit back and complain). All in all, it would probably be better just to interlink the two (concavity and convexity) religiously -- say a link to concavity in the main definition. This is a good page right now: concise and exactly what people are looking for. Don't see much need to change it. —The preceding unsigned comment was added by Semanticprecision (talk • contribs) .
: OK, I did not do any merge, rather moved concavity (which is now disambig) to concave function. Things might still need merging in the future, but at least for now the meaning of concave set and concave function are separate and not in the same article (with concave function taking the lion share of the room). Let us see how it goes. Oleg Alexandrov (talk) 00:28, 30 December 2005 (UTC)
replies: text: In February 2005, I merged convex and concave function in the German Wikipedia (:de:konvexe und konkave Funktionen). IMHO it is very difficult to maintain the versions for convex/concave funtion in a consistent way, and it is easy to write the article in not too confusing way. To avoid problems like in supermodular, clearly the title should mention both (e.g. convex and concave functions, as in the German version).  --NeoUrfahraner 09:15, 2 January 2006 (UTC)
thread_title:  Merge with concavity? 

message: The link from "opposite" ("The opposite of a convex function is a concave function") to the additive inverses page seems a bit gratuitous. I'm removing it. --Dchudz 14:13, 25 July 2006 (UTC)
thread_title:  Removing link to Additive Inverse 

message: I didn't want to modify the article, but there is something suspicious to me.
It says:
A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. If its second derivative is positive then it is strictly convex, but the converse does not hold, as shown by f(x) = x4. 
More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set.
Shouldn't the Hessian reduce to 2nd derivative in one-dimensional case?
The example function f(x) = x4 is convex and its second derivative 12x2 is nonnegative, so I don't see why the converse does not hold. Moreover in the previous sentence it says "if and only if", so the converse MUST hold. Did I miss something ?  —Preceding unsigned comment added by 83.37.136.53 (talk) 09:55, 18 January 2008 (UTC)
replies: text:  As far as I see, the article says that if the second derivative is strictly positive, then the function is strictly convex, but if the function is strictly convex, its second derivative need not be strictly positive, with x^4 being a counterexample (this function is indeed strictly convex, but the second derivative is not strictly positive, it also takes the value 0). Oleg Alexandrov (talk) 15:36, 18 January 2008 (UTC)
thread_title:  The example with f(x)=x<sup>4</sup> 

message: what is "semi-strictly convext" of a function? Jackzhp (talk) 02:09, 30 November 2008 (UTC)
thread_title: semi-strictly convex

message: It would be good to have a picture of a convex decreasing function as well. See Indifference curve.
Trogsworth (talk) 19:12, 5 May 2009 (UTC)
thread_title:  Convex Decreasing 

message: It'd be really wonderful if along with some examples we could supply some interesting counterexamples.  —Preceding unsigned comment added by 68.65.169.203 (talk) 03:16, 17 February 2010 (UTC)
replies: text: I agree. In particular, it would be great if one of the first pictures showed a typical example of a convex and a non-convex function (all 1d of course) side by side, so that people can get a general idea just by looking at that and be done with it (if that was all they needed, which it often is). --92.209.35.171 (talk) 15:12, 17 October 2020 (UTC)
thread_title:  Counterexamples 

message: The article says:
Suppose ƒ is a one variable function defined on an interval, and let
: R(x,y) = \frac{f(x) - f(y)}{x - y}
(note that R(x,y) is the slope of the red line in the above drawing;  note also that the function R is symmetric in x,y). ƒ is convex if and only if R(x,y) is monotonically non-decreasing in x, for y fixed (or viceversa). This characterization of convexity is quite useful to prove the following results.
I have a problem with that: Consider the function  f(x)=x^3 . Its slope is R(x,y) = x^2 + xy + y^2. Consider the slope for y=2 fixed.
Then we have R(x,2)=x^2+2x+4. On the interval [-1,\infty[ this function is monotonically non-decreasing. However, ƒ is not
convex on that interval.
HermanMansta (talk) 23:42, 19 May 2010 (UTC)
thread_title:  Problem 

message: Regarding telling whether a function is strictly convex by looking at its 2nd derivative, it's interesting that the only counter-example ( f(x) = x^4 ) is one where  \, f(x)  is only zero for a single value of  x .  I would speculate that for any example of a differentiable function that is strictly convex but doesn't have a strictly positive 2nd-derivative, that its 2nd derivative would still be positive almost everywhere.
I'm just speculating, but would it be correct to make the following change?
Current wording:
: f\,\! strictly convex if  f(x) > 0 \,\! for all x\,\! (note: this is sufficient, but not necessary)
I'd suggest:
: f\,\! strictly convex if and only if  f(x) \ge 0  for all x\,\! and the set  \{x \, : \, f(x)=0\}  has zero Lebesgue measure.  —Preceding unsigned comment added by Bleachpuppy (talk • contribs) 22:07, 30 August 2010 (UTC)
replies: text:  Technically, a convex function doesn't have to have second derivatives everywhere; all we know is that the first derivative exists almost everywhere and is increasing (for convex) and strictly increasing (for strictly convex).  It's likely you're correct about the second derivative condition; however, since it's not obvious, we need a reference.  — Arthur Rubin  (talk) 23:02, 30 August 2010 (UTC)
replies: text: Look at Rademacher's theorem, which is proved in L.C. Evan's book on fine properties of real functions and probably in Francis Clarke's (SIAM-reprinted) book on nonsmooth analysis. Jeff Cheeger gave an interesting proof some years ago, which is especially interesting for metric measure spaces. Older references include Busemann's  book on convex surfaces.  Kiefer.Wolfowitz  (talk) 20:47, 7 February 2011 (UTC)
thread_title:  necessary conditions on 2nd derivative for strict convexity 

message: i want to discuss a simple topic related to convex function  —Preceding unsigned comment added by 124.124.247.13 (talk) 19:17, 7 February 2011 (UTC)
thread_title:  ashutosh yadav 

message: Extended real-valued functions are useful for representing life-and-death constraints: Their violations receive the value of positive infinity (which is not a minimum for most erv functions). This is the keypoint, which is now absent from the article.
Please follow the terms of Rockafellar and Wets or the latest book by Borwein and Vanderwerff: There is no need for OR proposing to reformulate the basic terms of an established subdiscipline.  Kiefer.Wolfowitz 18:57, 25 April 2011 (UTC)
thread_title: Optimization and properness

message: The start of article is confusing "joining any two points of the graph; or more generally, any two points in a vector space." what is any two points? requires more explanation.Osmankhalid2005 (talk) 18:05, 29 January 2013 (UTC)
thread_title: Joining any two points

message: The letter "f" (denoted function) needs to have a space before the successor word. what doesn't happen here. Please fix it up :) 46.19.85.89 (talk) 15:06, 17 August 2014 (UTC)
replies: text: Can you give an example?  — Arthur Rubin  (talk) 14:45, 21 August 2014 (UTC)
thread_title:  spaces between words 

message: Please explain the history of this term, and why such graphs are called convex (viewed from below) and not concave (viewed from above). 
—DIV (120.17.118.20 (talk) 13:29, 8 October 2018 (UTC)
replies: text: I think historical overview belongs to Convex optimisation. AVM2019 (talk) 04:21, 25 September 2020 (UTC)
thread_title:  History/Origin 

message: What is the import of "in a Euclidean space (or more generally a vector space) of at least two dimensions" in the first sentence?  I don't find anything like that mentioned elsewhere in the article. McKay (talk) 06:21, 18 January 2019 (UTC)
thread_title:  Obscure leading sentence 

message: An editor has asked for a discussion to address the redirect Strictly convex. Please participate in the redirect discussion if you wish to do so.  1234qwer1234qwer4 (talk) 18:50, 21 April 2020 (UTC)
thread_title:  "Strictly convex" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: One of the concave function mentions is wrong and must be replaced with convex.
Since I am not familiar with these terms someone who knows them from those calculus books mentioned should do the edit.
--Christian Brech (talk) 21:18, 24 August 2020 (UTC)
thread_title:  Convex down & Convex up 

message: There is a powerful generalization of convexity to matrices called *operator convexity*, and there are at least two related variations of Jensen's inequality (the trace Jensen inequality and the operator Jensen inequality) for matrices.  It would be nice if there were some link to it or mention of it in this article.  (A search engine directed me to this page after searching for "integral representations of operator convex functions", and there isn't a link to head in the right direction.) 60.250.123.226 (talk) 17:42, 8 July 2023 (UTC)
thread_title:  Needs link to operator convexity 

message: In the Properties Section, Functions of one variable, 1st bullet point, there is an unclear reference to a drawing. Recommend the wording 'is the slope of the purple line in the above drawing' be changed to include a reference to the specific image (it is the first image) or the article be reformatted to include the image nearer to the reference.
Fantasticawesome (talk) 19:05, 21 March 2024 (UTC)
thread_title:  Unclear reference to drawing/image