Talk:Convex function

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Mathematics rating:	Start Class	Mid Priority	Field: Analysis
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 22:49, 16 June 2007 (UTC)

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.

For me at least, this page is exactly what I was looking for about convex functions.

Contents 1 Correct Definition 2 Merge with concavity? 3 Removing link to Additive Inverse 4 The example with f(x)=x4

[edit] Correct Definition

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)

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!

[edit] Merge with concavity?

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)

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)

[edit] Removing link to Additive Inverse

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)

[edit] The example with f(x)=x⁴

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)

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 x⁴ 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)