Talk:Logarithmically concave function

Category

I wouldn't say this entry belongs to the Category Elementary algebra. Logarithmically convex function is in Category:Real analysis.--Kompik 17:27, 5 June 2006 (UTC)

Range

The statement

Examples of log-concave functions are the indicator functions of convex sets.

in the current version of the article seems incorrect to me -- the logarithm of a function which attains zero value is not defined. Or have I missed something -- perhaps the indicator function was meant in some other sens? --Kompik 20:55, 28 January 2007 (UTC)

The confusion stems from the apparent requirement that log-concave function be positive. This can be relaxed, with some care. I have attempted to provide the additional material necessary to support the relaxed definition in the new text. Mcgrant (talk) 17:41, 22 March 2011 (UTC)

I'm not at all into real analysis, but it seems to me that recent edits have made the article less coherent by removing the requirement of (even weakly) positive values. If there are negative values, their logarithm is not a real number, nor are their non-integer powers defined, so none of the introductory stuff seems to make sense. Note the presence in the current article of "It is often convenient to allow f to assume zero values as well, as long as the set of points for which f is positive remains a convex set" (which does not seem to consider negative values at all) and "Every concave function is log-concave", which also seems naive about negative values. Marc van Leeuwen (talk) 13:15, 23 March 2011 (UTC)

Fair point on "every concave function"; I will add "nonnegative". Also, Marc, I hope you will head over to logarithmically concave sequence and ensure that I didn't screw up your writing there. Mcgrant (talk) 17:54, 24 March 2011 (UTC)

That's all right, at first I was somewhat surprised by the disappearance of their mention here, but a separate article is OK. However to get back to this article, I really think it is confusing to start with

In mathematics, a positive function f : Rⁿ → R is logarithmically concave...

since the explicitly specified range R made me gloss over the "positive" just before. I would say, with a notation that is not standard but cannot be misconstrued:

In mathematics, a function f : Rⁿ → R_≥0 is logarithmically concave...

(I don't know why people stick to ambiguous things like "positive" (which means ≥0 here in France but >0 in English texts) or R⁺ (could very well be the additive group of the reals).) But seriously, whatever notation is used, make it coherent. I understand that the article is now about functions to the non-negative real numbers, but it doesn't say that. Marc van Leeuwen (talk) 18:13, 24 March 2011 (UTC)

I'll keep working on it. I sincerely thank you for the criticisms. Frankly if you look at the original source (Boyd & Vandenberghe 2004) there's a lot of meat there that Wikipedia just doesn't yet have. For instance, a proper treatment of the subject needs to address the notion of functions with limited domains (I've taken a crack at that here), and Wikipedia should probably have a page on extended-valued functions. Mcgrant (talk) 18:23, 24 March 2011 (UTC)

Aha! I think that if I delay the actual discussion of the logarithm until the second paragraph, I can guide the discussion a bit better and address some of your concerns. Stay tuned. Mcgrant (talk) 18:30, 24 March 2011 (UTC)

OK. Let me know what you think now. What I've done is define log-concavity strictly in terms of the inequality, which applies without modification to functions that obtain zero values. *Then*, I say that if the function is strictly positive, this is equivalent to saying that the logarithm of the function is concave. This avoids having to even introduce the ideas of extended-valued functions, the logarithm of zero, and so forth. Mcgrant (talk) 18:46, 24 March 2011 (UTC)