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I'm not sure what the opinion at Wikipedia is on the matter of accessibility of the material. When I come here, however, I am not looking for mathematically concise definitions of what I look up, nor do I handle well the high level description that requires knowledge of many other technical terms.
A paragraph describing the significance of the defining equation would be nice.
Should the definition read "for all x,y" instead of for any?
But anyone writing math would never say 'for any', as strictly this requires the condition to be satisfied just for some, and not for all instances. In this article, I am missing the 'useful theorems' section for higher dimensions, as well as a generally better treatment for the high dimensions. But, I still appreciate your work!
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) .
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)
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)
Nice! what if the second derivative is f``(x)=A(x-x1)^2(x-x2)^2 ? The second derivative is zero at 2 points. "f(x) defined on an interval is called convex (or convex downward or concave upward) if the graph of the function lies below the line segment joining any two points of the graph." ... it should be on or below a line? you could have a continuous set of points where f(x) is zero. -Alok 09:25, 30 August 2012 (UTC)
what is "semi-strictly convext" of a function? Jackzhp ( talk) 02:09, 30 November 2008 (UTC)
It would be great to add a section with the definition of a strongly convex function; or, make a page for "strongly convex" (since it appears that it doesn't yet exist) and then link to that page. 71.130.221.31 ( talk) 02:46, 2 February 2009 (UTC)
Hi folks. I'm wondering if there's a typo in the final example in the "Strongly convex functions" section. Shouldn't be replaced by in the sentence starting "For example, consider a function that is strictly convex"? 67.186.29.135 ( talk) 18:15, 2 November 2012 (UTC)
Hi, I have added a useful result: "A twice continuously differentiable function f on a compact domain that satisfies for all is strongly convex. The proof of this statement follows from the well-known Weierstrass theorem, which states that a continuous function on a compact set has a maximum and minimum." Surprisingly I cannot find this result in any book. The proof is straightforward although I can formalize it if needed. Erikjm ( talk) 20:23, 13 August 2014 (UTC)
I was just told that an equivalent definition is the following: a function is convex if at any point we can find a suppporting line (a line through the point that lies entirely below the function), i.e. f is convex if for any a, there exists an m such that for all x. (It's easy to see why if that condition is true the function is convex, but I can't immediately think of a proof for the other direction.) Wondering if someone knows a source.... I could only find Springer EOM, which indicates that maybe this definition is equivalent to midpoint convex. Shreevatsa ( talk) 01:15, 18 February 2009 (UTC)
This is somehow related but have anyone see this equivalent definition, something like: a function is convex if and only if for any arbitrary E
where I don't remember what E should be :) a compact set?. The above is incorrect in the way it is written; I don't remember the exact formulation. I also remember the converse of Jensen inequality is true (under some conditions). (This was an exercise in Rudin's real and complex analysis.) So, it should give another equivalent definition. -- Taku ( talk) 13:42, 18 February 2009 (UTC)
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)
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)
The article says:
Suppose ƒ is a one variable function defined on an interval, and let
(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 . Its slope is . Consider the slope for y=2 fixed. Then we have . On the interval this function is monotonically non-decreasing. However, ƒ is not convex on that interval.
HermanMansta ( talk) 23:42, 19 May 2010 (UTC)
Regarding telling whether a function is strictly convex by looking at its 2nd derivative, it's interesting that the only counter-example () is one where is only zero for a single value of . 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:
I'd suggest:
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)
I'm proposing that Proper convex function be merged into Convex function. If " proper convex function" is at all notable, it should be noted that a "proper convex function" on ℝn corresponds exactly to a convex function on a convex subset of ℝn. Some of the properties remain the same, although the infimal convolute is simpler to define for total functions to the extended real line than for partial functions to the real line. — Arthur Rubin (talk) 15:44, 11 March 2011 (UTC)
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)
KW's edit summary in reverting my edits to the lead was: "convex functions need not be defined on intervals, but frequently are defined on higher dimensional sets". But this certainly suggests that he had not read past the first two sentences of the lead, where it says "More generally, this definition of convex functions makes sense for functions defined on a convex subset of any vector space." I have also moved the more equation-based definition of convexity to a "Definition" section, with some copyedits. We should keep WP:LEAD and WP:MTAA in mind. I don't see any reason that the lead should contain any equations at all. Sławomir Biały ( talk) 22:15, 24 April 2011 (UTC)
Moved from User talk:Sławomir Biały:
The epigraph definition also remains in the first paragraph of the lead. Also, I'm not really sure that "concave up" means the same thing as convexity. This is a pointwise property, that the graph of a function locally lies over the tangent hyperplane, which is slightly different. There is, of course, a strong connection between this and convexity, and in many cases of interest they are the same, but I don't think that the two should be equated in the article. Any thoughts on this? Sławomir Biały ( talk) 22:28, 24 April 2011 (UTC)
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)
In the section about the properties of a convex function it is said that if f is contiuous on some open interval, then it has one sided derivatives. As a consequence, it's differentiable at all but at most countably many points. How is this conclusion inferred from the one sided derivaatives? And what happens if f is convex on closed interval? Million of thanks for answerring these questions! :) 46.19.85.157 ( talk) 10:27, 11 February 2014 (UTC) ??? 132.66.85.52 ( talk) —Preceding undated comment added 09:30, 17 February 2014 (UTC)
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)
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))
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)
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)
In introductory level math books, the term convex is often conflated with the opposite term concave by referring to a "concave function" as "convex downward". Likewise, a "concave" function is referred to as "convex upwards" to distinguish it from "convex downwards".
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)
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)
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)
![]() | This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
|
I'm not sure what the opinion at Wikipedia is on the matter of accessibility of the material. When I come here, however, I am not looking for mathematically concise definitions of what I look up, nor do I handle well the high level description that requires knowledge of many other technical terms.
A paragraph describing the significance of the defining equation would be nice.
Should the definition read "for all x,y" instead of for any?
But anyone writing math would never say 'for any', as strictly this requires the condition to be satisfied just for some, and not for all instances. In this article, I am missing the 'useful theorems' section for higher dimensions, as well as a generally better treatment for the high dimensions. But, I still appreciate your work!
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) .
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)
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)
Nice! what if the second derivative is f``(x)=A(x-x1)^2(x-x2)^2 ? The second derivative is zero at 2 points. "f(x) defined on an interval is called convex (or convex downward or concave upward) if the graph of the function lies below the line segment joining any two points of the graph." ... it should be on or below a line? you could have a continuous set of points where f(x) is zero. -Alok 09:25, 30 August 2012 (UTC)
what is "semi-strictly convext" of a function? Jackzhp ( talk) 02:09, 30 November 2008 (UTC)
It would be great to add a section with the definition of a strongly convex function; or, make a page for "strongly convex" (since it appears that it doesn't yet exist) and then link to that page. 71.130.221.31 ( talk) 02:46, 2 February 2009 (UTC)
Hi folks. I'm wondering if there's a typo in the final example in the "Strongly convex functions" section. Shouldn't be replaced by in the sentence starting "For example, consider a function that is strictly convex"? 67.186.29.135 ( talk) 18:15, 2 November 2012 (UTC)
Hi, I have added a useful result: "A twice continuously differentiable function f on a compact domain that satisfies for all is strongly convex. The proof of this statement follows from the well-known Weierstrass theorem, which states that a continuous function on a compact set has a maximum and minimum." Surprisingly I cannot find this result in any book. The proof is straightforward although I can formalize it if needed. Erikjm ( talk) 20:23, 13 August 2014 (UTC)
I was just told that an equivalent definition is the following: a function is convex if at any point we can find a suppporting line (a line through the point that lies entirely below the function), i.e. f is convex if for any a, there exists an m such that for all x. (It's easy to see why if that condition is true the function is convex, but I can't immediately think of a proof for the other direction.) Wondering if someone knows a source.... I could only find Springer EOM, which indicates that maybe this definition is equivalent to midpoint convex. Shreevatsa ( talk) 01:15, 18 February 2009 (UTC)
This is somehow related but have anyone see this equivalent definition, something like: a function is convex if and only if for any arbitrary E
where I don't remember what E should be :) a compact set?. The above is incorrect in the way it is written; I don't remember the exact formulation. I also remember the converse of Jensen inequality is true (under some conditions). (This was an exercise in Rudin's real and complex analysis.) So, it should give another equivalent definition. -- Taku ( talk) 13:42, 18 February 2009 (UTC)
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)
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)
The article says:
Suppose ƒ is a one variable function defined on an interval, and let
(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 . Its slope is . Consider the slope for y=2 fixed. Then we have . On the interval this function is monotonically non-decreasing. However, ƒ is not convex on that interval.
HermanMansta ( talk) 23:42, 19 May 2010 (UTC)
Regarding telling whether a function is strictly convex by looking at its 2nd derivative, it's interesting that the only counter-example () is one where is only zero for a single value of . 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:
I'd suggest:
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)
I'm proposing that Proper convex function be merged into Convex function. If " proper convex function" is at all notable, it should be noted that a "proper convex function" on ℝn corresponds exactly to a convex function on a convex subset of ℝn. Some of the properties remain the same, although the infimal convolute is simpler to define for total functions to the extended real line than for partial functions to the real line. — Arthur Rubin (talk) 15:44, 11 March 2011 (UTC)
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)
KW's edit summary in reverting my edits to the lead was: "convex functions need not be defined on intervals, but frequently are defined on higher dimensional sets". But this certainly suggests that he had not read past the first two sentences of the lead, where it says "More generally, this definition of convex functions makes sense for functions defined on a convex subset of any vector space." I have also moved the more equation-based definition of convexity to a "Definition" section, with some copyedits. We should keep WP:LEAD and WP:MTAA in mind. I don't see any reason that the lead should contain any equations at all. Sławomir Biały ( talk) 22:15, 24 April 2011 (UTC)
Moved from User talk:Sławomir Biały:
The epigraph definition also remains in the first paragraph of the lead. Also, I'm not really sure that "concave up" means the same thing as convexity. This is a pointwise property, that the graph of a function locally lies over the tangent hyperplane, which is slightly different. There is, of course, a strong connection between this and convexity, and in many cases of interest they are the same, but I don't think that the two should be equated in the article. Any thoughts on this? Sławomir Biały ( talk) 22:28, 24 April 2011 (UTC)
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)
In the section about the properties of a convex function it is said that if f is contiuous on some open interval, then it has one sided derivatives. As a consequence, it's differentiable at all but at most countably many points. How is this conclusion inferred from the one sided derivaatives? And what happens if f is convex on closed interval? Million of thanks for answerring these questions! :) 46.19.85.157 ( talk) 10:27, 11 February 2014 (UTC) ??? 132.66.85.52 ( talk) —Preceding undated comment added 09:30, 17 February 2014 (UTC)
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)
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))
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)
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)
In introductory level math books, the term convex is often conflated with the opposite term concave by referring to a "concave function" as "convex downward". Likewise, a "concave" function is referred to as "convex upwards" to distinguish it from "convex downwards".
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)
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)
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)