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This page states that a twice-differentiable function is concave if and only if (iff) the second derivative is negative. I don't think this is correct.

A twice-differentiable function is STRICT concave if and only if the second derivative is negative.

Use a constant function as an example. By the simpler definition, a constant function (or a linear function) is always concave (but not strictly concave). However, the second derivative is ZERO, not negative.

You are right; I corrected the text. --NeoUrfahraner 14:17, 1 Mar 2005 (UTC)

But what about;

A differentiable function f is concave on an interval if and only if its derivative function f ′ is monotonically decreasing on that interval, that is, ${\displaystyle f''<0}$: a concave function has a decreasing slope.

Shouldn't this be ${\displaystyle f''\leq 0}$ too (+ the slope can be constant)? — Preceding unsigned comment added by Bgst (talk • contribs) 18:16, 23 August 2017 (UTC)Reply

I agree. I just changed it accordingly. Does it look better now? Yak90 (talk) 16:14, 10 October 2017 (UTC)Reply

I think the "if and only if" for strict concavity needs to be relaxed though. Or the "negative" should be changed to "negative definite" or something similar.

In other words, take -x^4. This function is clearly STRICT concave. However, its second derivative AT THE ORIGIN is 0. Thus, a function is strict concave if its second derivative is less than or equal to zero with EQUALITY ONLY at the origin. That sounds like negative definite. <?>

If you use the "definiteness" terms, I think this extends well into multiple variables as well.

Just a thought.

OK, corrected again --NeoUrfahraner 11:14, 2 Mar 2005 (UTC)

Merging looks like a good idea to me, but one should merge this article in convex function and not viceversa I think. Also, convex function looks like a rather well-written article, so hopefully the merged version will not be worse than what it is now. In short, if anybody is willing to merge, that person should be willing to take the necessary time to do a good job. Otherwise I would oppose a merger. Oleg Alexandrov (talk) 12:12, 24 October 2005 (UTC)Reply

See also Talk:Convex function. Oleg Alexandrov (talk) 00:29, 30 December 2005 (UTC)Reply

I agree with this, it is analogous to having monotone function an article for increasing and one for decreasing. --Santropedro (talk) 00:12, 19 June 2017 (UTC)Reply

"The definitions of convex and concave functions given here appear to be incorrect. What is said to be a concave function is a convex function and what is said to be a convex function is a concave function.

I have always found those terms confusing. The Chinese characters for the two terms have the shapes that they describe, so it is easy to recognise. The charactor for convex (or concave downward) is a protrusion on top of a square, and the one for concave (or concave upward) is a depression on the top side of a square."

Sorry it seems that I posted the above by mistake. I pressed "Save" and did not realise that my comment is already posted. I did not mean to edit the page but only wanted to ask the author the page to check carefully. I have not learned how to delete the paragraphs I added at the end of the item, which look ugly. Could you delete them? Please check your definitions.

Ziheng Yang

Would you mind making an account if you would like to contribute more? It is better to keep in touch that way.

As far as I can tell, everything is correct. Concave up is the same as convex, meanining a function of this form" \_/

Concave down is the same as concave, meaning a function of this form:

  ----
 /    \
/      \

Other comments? Oleg Alexandrov (talk) 18:27, 30 December 2005 (UTC)Reply

@172.189.190.112: (Ziheng Yang) It is the Chinese knowledge that makes you misunderstand. As a native Chinese speaker, I used to have the same misunderstanding too. The English term are not exactly the same as the Chinese term. The English term "concave" means "curving in", but the Chinese term 凹 primarily means "lower than the ground" and secondarily means "curving in". The English term "convex" means "curving out", but the Chinese term 凸 primarily means "higher than the ground" and secondarily means "curving out". When we see the English term "concave function" and translate it into Chinese 凹函数, we usually take the primary meaning in Chinese so the literal meaning seems to be the "function lower than the ground", like this \/, but it really does not mean this in English. The English term literally means the function curving in (relative to the upper area of the function curve), so it should look like this /\. --Yejianfei (talk) 12:23, 2 March 2020 (UTC)Reply

Check use of convex/concave with "concave lens" and "convex lens", convex "bulges out", concave "caves in", isn't that the correct definition? Why is use with function different?

What happens when you apply this definition to linear functions? This article does not make mention of this case. --Stux 17:34, 16 February 2006 (UTC)Reply

What is the problem? Linear functions are concave, but not strictly concave. --NeoUrfahraner 18:45, 16 February 2006 (UTC)Reply

Ok I sort of see it being sense mentioned here, at least in terms of linear transformation functions, but nowhere is it explicitly mentioned as such and briefly explained. I, personally am still unclear as to why it's either (I can see that if it's convex it'll be concave too), since this article stipulates:

If f(x) is twice-differentiable, then f(x) is concave iff f ′′(x) is non-positive.

Linear functions are clearly twice-differentiable and f''(x) = 0, of course, and this is ... Oh I just answered my own question! Nevermind, Thanks! But I still think it needs to be more explicitly mentioned in these articles. --Stux 22:36, 16 February 2006 (UTC)Reply

I think that the definition is obvious enough to show that linear functions are concave.--A 17:07, 13 October 2007 (UTC)Reply

"Equivalently, f(x) is concave on [a, b] if and only if the function −f(x) is convex on every subinterval of [a, b]."

I feel like this line might be glomming together two separate points. Are both of the following correct?:

1) A function f is concave an [a,b] iff -f is convex on [a,b].

2) A function f is concave (convex) on [a,b] iff it is concave (convex) on every /proper/ (was this word left out of the original?--it makes the statement less trivial) subinterval of [a,b].

If these are both true, maybe they should replace the current language. Also, the placement of this part suggests that the statement is restricted to continuous functions. Is that desired? --Dchudz 15:48, 25 July 2006 (UTC)Reply

That is right, I think it should read "Equivalently, f(x) is concave on [a, b] if and only if the function −f(x) is convex on [a, b]"--A 19:24, 13 October 2007 (UTC)Reply

The current definition uses a concave set. This makes no sense. Both convex and concave functions are defined over CONVEX sets. I'm going to change it to a convex set.

http://www.chass.utoronto.ca/~osborne/MathTutorial/CVN.HTM

--TedPavlic 17:44, 11 May 2007 (UTC)Reply

---

{{technical}}

69.140.152.55 (talk) 16:28, 19 May 2008 (UTC)Reply

I have removed the {{technical}} tag, as the article now meets WP:MTAA. Gandalf61 (talk) 16:06, 13 August 2008 (UTC)Reply

This article could benefit from a few simple examples. zermalo (talk) 18:29, 20 May 2008 (UTC)Reply

I added a few. Oleg Alexandrov (talk) 04:02, 21 May 2008 (UTC)Reply

I added a link to a not very common application. It is found at the end of the section on boundary value problem of the article Computation of radiowave attenuation in the atmosphere --Thuytnguyen48 (talk) 14:57, 18 October 2010 (UTC)Reply

A graph, similar to the one for Convex function would be useful here. —Preceding unsigned comment added by Nandhp (talk • contribs) 15:14, 2 November 2008 (UTC)Reply

I would say at the beginning of this article, we should put something like this:

• quasiconcave: ${\displaystyle f\left(tx_{1}+\left(1-t\right)x_{2}\right)\geq \min \left\{f\left(x_{1}\right),f\left(x_{2}\right)\right\}}$
• strictly quasiconcave: ${\displaystyle f\left(tx_{1}+\left(1-t\right)x_{2}\right)>\min \left\{f\left(x_{1}\right),f\left(x_{2}\right)\right\},\forall x_{1}\neq x_{2}}$
• weakly concave: ${\displaystyle f\left(tx_{1}+\left(1-t\right)x_{2}\right)\geq tf\left(x_{1}\right)+\left(1-t\right)f\left(x_{2}\right)}$
• strictly concave: ${\displaystyle f\left(tx_{1}+\left(1-t\right)x_{2}\right)>tf\left(x_{1}\right)+\left(1-t\right)f\left(x_{2}\right),\forall x_{1}\neq x_{2}}$

When people don't say it clearly, it means weakly concave.

what fields use quasiconcavity? How related is it to concavity? 0¹⁸ (talk) 02:35, 18 August 2010 (UTC)Reply

Hi, nice for the response, what field? you can read the quasiconcave page. I don't really know since this is about math. By the way, do you have any idea what fields use the "usual concavity"? Jackzhp (talk) 13:20, 18 August 2010 (UTC)Reply

What I'm asking you is this: why do you want to include the concept of quasiconcavity? how is it related to concavity? I'd argue a see also link would probably make sense. The basic point is someone interested in concavity is very likely not interested in quasiconcavity. 0¹⁸ (talk) 17:11, 18 August 2010 (UTC)Reply

O. I feel that I still didn't get your point. Do you mean quasiconcavity has nothing to do with the concavity? For me, I feel that if the article doesn't mention quasiconcave or it doesn't put the way I proposed, it missed something. And I thought concavity has 3 extent: quasi, weak, strong, isn't it? I thought they are related and I was very easily confused by them, especially when they were put far away. For me, the above 3 lines say everything one needs, we don't even need the 2 whole articles.

If I understand the terms correctly, it seems that you are saying it is appropriate in the "human" article, we only mention "women", give only a link to "man". and you are saying "someone interested in" human "is very likely not interested in" man. Am I very wrong? I am laughing for the situation. Jackzhp (talk) 18:19, 18 August 2010 (UTC)Reply

I noticed that when viewed in Safari on my iPhone, the definition of concavity shows up incorrectly: the term \alpha f(y) does not appear. The page looks fine in multiple browsers on my laptop, and I don't see what could be causing the problem.Danramras (talk) 22:34, 6 September 2017 (UTC)Reply

Please explain the history of this term, and why such graphs are called "concave" (viewed from below) and not "convex" (viewed from above). —DIV (120.17.118.20 (talk) 13:31, 8 October 2018 (UTC))Reply

Pretty sure the equations in the definition section are wrong. Leopd (talk) 08:16, 31 January 2021 (UTC)Reply

Actually, the equations are correct, but the layout on the iOS Wikipedia app is very confusing. Due to the limited width of the screen, when viewing the page vertically, the first equation reads as

${\displaystyle f((1-\alpha )x+\alpha y)\geq (1-\alpha )}$

instead of

${\displaystyle f((1-\alpha )x+\alpha y)\geq (1-\alpha )f(x)+\alpha f(y)}$

Confusingly, if you scroll the page sideways to see more content to the right, you don't see the rest of the equation. You see a bunch of blank space to the right of all the text, and the equation seems to be complete. The only way to see the rest of the equation is to scroll just in the equation box itself. Unless you happen to drag your finger horizontally on the equation, there is no visual affordance that there is anything more to the equation.

This is not an issue with the page so much as the iOS client. Where would that topic be best brought up? Leopd (talk) 20:32, 2 February 2021 (UTC)Reply