Wikipedia:Featured article candidates/Shapley–Folkman lemma/archive1

The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article candidates. No further edits should be made to this page.

The article was not promoted by Karanacs 01:41, 13 October 2011 [1].

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Shapley–Folkman lemma

Shapley–Folkman lemma (edit | talk | history | protect | delete | links | watch | logs | views)

• Featured article candidates/Shapley–Folkman lemma/archive1

Nominator(s):  Kiefer.Wolfowitz 00:40, 24 September 2011 (UTC)

I am nominating this for featured article because... it is a comprehensive, well-documented, clearly written article on the Shapley-Folkman lemma and its applications and because it features two graphs (created by User:David Eppstein).  Kiefer.Wolfowitz 00:40, 24 September 2011 (UTC)

• The Nobel Prize in Economics shall be awarded on 10 October after 1:00 p.m. CET (Monday). It would be desirable to feature this article on the day on which the Nobel prize is awarded 10 October 2011 or in 2012.  Kiefer.Wolfowitz 05:35, 1 October 2011 (UTC)

Copyscape review - No issues were revealed by Copyscape searches. Graham Colm (talk) 14:32, 24 September 2011 (UTC)

Thanks! :)  Kiefer.Wolfowitz 20:06, 30 September 2011 (UTC)

Provisional support - I was involved in the A class review and approved of the article generally then. However the article has been expanded significantly since then. I will take a close look at the applications section but I don't feel comfortable with the rest of the text. Protonk (talk) 00:07, 28 September 2011 (UTC)

• The Applications section covers three (somewhat) conceptually distinct subjects; economics, optimization and probability theory. The text introducing the section should give the reader a mini road map of what to expect from the subordinate parts.

  True. I believe that the OR by synthesis is trivial: I am synthesizing statements made within 3 disciplines, which are obviously true, and saying that the statement is generally true, and has giving three examples of the statement.

  UPDATE: "The Shapley–Folkman lemma enables researchers to extend results for Minkowski sums of convex sets to sums of general sets, which need not been convex. Such sums of sets arise in economics, in mathematical optimization, and in probability theory; in each of these three mathematical sciences, non-convexity is an important feature of applications and the Shapley–Folkman lemma has renewed research that had been stumped by non-convex sets. In all three disciplines, the break-through application of the Shapley–Folkman lemma has been made by a young scientist" (whose innovations have then spread through the discipline ...).

  I have warned about possible minor OR by synthesis. I also want to inspire the youth to unleash their barbaric YAWP (like Lemarechal or Ekeland/Aubin or Starr or Artsein/Vitale---or Galois or Thomas Paine or Tom Kahn or The Kinks and the The Replacements).  Kiefer.Wolfowitz 03:17, 1 October 2011 (UTC)

  Conceivably, I could be asked to provide citations establishing all the authors' youth: If so, then I'd just delete the inspirational statements about young researchers shaking up things. K.W.

  I don't think synthesis is necessary. Right now we have a one sentence introduction for the three sections. I am suggesting a few sentences more (some of which could be recapitulated later on) to give the reader a quick indication of what is to come. Protonk (talk) 17:37, 3 October 2011 (UTC)

  Perhaps you have not seen the applications-introduction after I updated it a few days ago? I provided a short overview. However, it is not possible to go into (much) more detail, because each application has a special vocabulary needing explanation.  Kiefer.Wolfowitz 12:29, 4 October 2011 (UTC)

  No I didn't see it. With the added few sentences it looks great. Protonk (talk) 19:51, 4 October 2011 (UTC)

• "On this set of baskets, an indifference curve is defined for each consumer..." the two paragraphs in Economics feel somewhat out of order. Given the context of this article, we don't need to resort to describing preferences as indifference curves before bringing to bear the topological intuition. I would introduce the concepts of baskets of goods and constraints. Then note that optimal choices among those goods under those constraints under the assumption of convexity leads to relatively simple and painless intuition and graphical explanation. Then segue into indifference curves. After that you can return to the last few sentences in the second paragraph. Now you are set up for explaining why non-convex preferences might need another solution.

  Using indifference curves allows us to work with convex sets. If we work with maximizing utility, then we have to introduce quasi-concave functions (having convex upper-levelsets), which is a more complicated approach, imho; if we discuss maximizing quasi-concave functions, then economics should follow optimization. (I first took this material from existing articles and then reworked it with precise references. I thought that our explanation was similar to Varian's Intermediate Economics, which I read 20 years ago: My memory has failed me before.) K.W.

  No, you're correct. Let me think about how to better explain my concern with this paragraph, which are almost entirely about ordering. Protonk (talk) 00:08, 1 October 2011 (UTC)

• The non-convex preferences section is ok, but I would like to see a more general explanation to begin with. You give a good example of non-connected demand (though it is basically copied from the summary article) and the Hotelling quote is handy. However you might want to open up a bit more generally or offer some examples from more well behaved areas where preferences are non-convex. If you prefer you could shorten the section a bit.

  Just 'okay'!?!!! ;) What does your "summary article" refer to? (Certainly not Ross M. Starr's New Palgrave survey, which has no examples of anything!) My example was similar to examples given from Wold, Morgenstern, etc., and inspired by Harry Potter's Hogwart's school and by memories of Dungeons and Dragons! :D (I remember Starr's Econometrica article having some nice examples.) This may be a matter of taste; students should use this while they are looking at an intermediate or M.A. textbook in microeconomics, and most of these have examples of non-convexities; more advanced students and researchers will find my literature collection (with references and precise page numbers) very valuable, I hope. I think that one example and pointers to good textbooks should suffice for the general public, particularly because we have an article on the convexity of preferences. K.W.

  I mean that section looks identical to Non-convexity (economics), both of which you wrote. I didn't mean to imply that you did anything malign. Protonk (talk) 00:11, 1 October 2011 (UTC)

  Okay, now I understand. :) I emboldened my wink above, and added a smiling face. :D Actually, the non-convex economy article was a spin-off from this. I realized that Sraffa and company's contributions to non-convexities and production economics discussed problems besides aggregation (and this article was becoming long).

• "The previously noted papers were listed" I'm not sure which papers you mean.

  The JPE, Shapley-Shubik, and Aumann papers were discussed by Starr's Econometrica paper. K.W.

  Improved: "Previous publications on non-convexity and economics were collected in an annotated bibliography by Kenneth Arrow. He gave the bibliography to Starr, who was then an undergraduate enrolled in Arrow's (graduate) advanced mathematical-economics course." K.W.

• I know I said this in the A class review but the article really doesn't need File:Price of market balance.gif.

  It is nice to show a demand function, because our consumer theory lacks a demand function. Showing an equilibrium can only help non-economists. (I agree that the illustration is not essential, and wouldn't object if another editor and you removed it.) K.W.

  Volunteer Marek also finds this graph sub-optimal, for another reason. The graph shows only one market, whereas our article discusses general equilibria (for N markets where N is a positive integer). However, N=1 is a special cases of general equilibria, so I discount VM's vote in this instance. Again, if anybody wants to remove the graph, then I would not object.  Kiefer.Wolfowitz 10:24, 2 October 2011 (UTC)

  DONE! I hid the graph (because 2>1).  Kiefer.Wolfowitz 15:44, 2 October 2011 (UTC)

• "Following Starr's 1969 paper..." This paragraph can be extended (at the expense of the non-convexity section if you wish). We get (or at least I get) the immediate implication for general equilibrium models but the article should offer some other more concrete examples.

  This may be a matter of taste. I think that the article is rather long (although much of the length is due to the meticulous referencing, which may turn off some readers); I am skeptical about the value of more applications. I would rather have more pictures, perhaps an animation of the set I mentioned on the talk page of the article (similar to Mas-Colell's example).  Kiefer.Wolfowitz 03:15, 1 October 2011 (UTC)

  On second thought, I agree that an example of another economic application would be useful. I suppose that the aggregation of non-convex budget sets would be interesting, and I am aware of estimated (non-convex) budget sets for Swedish consumers.

  However, I do not know of any public-domain graphics for single consumers. I do not have access to my library for 2 weeks, but the references I gave were all rather mathematical. I am afraid that it may be difficult to give an empirical example of an application of the SF lemma (using a real-world agent's estimated budget set or production set or preferences) without doing OR. K.W.

  This suggestion is good, but it lies beyond the scope of this FA nomination, imho. Do you agree?  Kiefer.Wolfowitz 15:48, 2 October 2011 (UTC)

• In that same paragraph you should remove the references to mathematical optimization and measure theory, as you are about to explain those in detail.

  I believe you are referring to Aubin's book on mathematical techniques for game theory and economics and then Trockel's book, which uses 2-3rd year Ph.D.-level mathematical analysis (ergodic theory, differential geometry/topology) to study economics. Aubin's book has a lot of non-probabilistic mathematics, and an extensive and original treatment of game theory and some economic models; I do cite it later in optimization, of course, because of the results with Ekeland. These two books (Aubin and Trockel) don't fit in the later sections (although your conjecture was very reasonable).  Kiefer.Wolfowitz 03:15, 1 October 2011 (UTC)

• Should the optimization section link to Knapsack problem?

  Lemarechal's problem was different. K.W.

• "An application of the Shapley–Folkman lemma represents..." Maybe this sentence should be followed by a quick non-technical explanation.

  DONE! "Of course, the given optimal-point is a sum of points in the graphs of the original summands and of a small number of convexified summands, by the Shapley–Folkman lemma." (I have not given a further non-technical explanation, because I'm not sure what that would be, and your "maybe" question is not an actionable complaint ....)  Kiefer.Wolfowitz 16:04, 2 October 2011 (UTC)

• " In 1973, the young mathematician Claude Lemaréchal was surprised by his success..." this is a neat bit of information. Perhaps some elaboration would help. Explain to the reader why Lemaréchal's results could be surprising without Shapley-Folkmann but are understandable with it. The text kind of does this now but it might help to make it more clear.

  Thanks!

  I wrote the following expansion:

  "In 1973, the young mathematician Claude Lemaréchal was surprised by his success with convex minimization methods on problems that were known to be non-convex; for minimizing nonlinear problems, a solution of the dual problem problem need not provide useful information for solving the primal problem, unless the primal problem be convex and satisfy a constraint qualification. Lemaréchal's problem was additively separable, and each summand function was non-convex; nonetheless, a solution to the dual problem provided a close approximation to the primal problem's optimal value. The crucial step in these publications is the use of the Shapley–Folkman lemma."

  With this edit, I believe that I have resolved every actionable problem. Sincerely,  Kiefer.Wolfowitz 12:56, 4 October 2011 (UTC)

• the probability section is dense, but good. I get the feeling that there is a deep connection between measure theory and Shapley-Folkmann but I can't pin it down.

  Searching for "Shapley Folkman" and "vector measure" on Google Scholar/Books will give you more food for thought. I thought that our treatment was appropriate for a summary style. Also, ending with the discussion by Vind, Debreu, Mas-Colell nicely ties the abstract mathematics with economics, imho.  Kiefer.Wolfowitz 19:49, 30 September 2011 (UTC)

Comments. I'm no mathematician, but I have a few observations nevertheless. This isn't particularly advanced mathematics, so we shouldn't be scared of it. (That was a rallying cry to other FA reviewers who may be as much math dunces as I am.)

• I think it's important that the lead is accessible to the general reader, who may not understand what a lemma is.

  New second sentence in lede: "In mathematics, lemmas are propositions that are steps in a proof of a theorem."  Kiefer.Wolfowitz 00:58, 28 September 2011 (UTC)

• "The Shapley–Folkman–Starr results address the question ...". No, they don't. Results don't address anything.

  "The Shapley–Folkman–Starr results provide an affirmative answer to the question, "Is the sum of many sets close to being convex?"  Kiefer.Wolfowitz 01:04, 28 September 2011 (UTC)

• "Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used to show that central results of (convex) economic theory are good approximations ...". What does "central results" mean?

  Updated"; for example, quasi-equilibria closely approximate equilibria of a convexified economy." (This rephrases a phrase only a few sentences earlier. I suspect that repetition may help readers rather than bore them.) K.W.

• "Minkowski addition is defined by the addition of the sets' members". Is it defined by it or as it?

  Sharply observed, MF! It should be "as". (Three corrections on page) K.W.

• "A real vector space of two dimensions can be given a Cartesian coordinate system in which every point is identified by a list of two real numbers". A list isn't two.

  I can substitute ordered pair. (DONE) K.W.

• "This distance is zero exactly when the sum is convex". What does that mean? Exactly zero when the sum is convex? Why "exactly"?

  "exactly" is a conversational way of writing "if and only if". I'll change it (because temporal "when" is distracting). (DONE) The word "exactly one" recurs in the image's alternative caption, because the statement that there exists 1 dollar in my bank account is true even when I have 2 or more dollars there. K.W.

• "The Shapley–Folkman–Starr theorem states that an upper bound on the distance between the Minkowski sum and its convex hull—the convex hull of the Minkowski sum is the smallest convex set that contains the Minkowski sum." Something's gone wrong with the punctuation or grammar there.

  That is weird. I'll check the history in case my keyboard mistyping deleted something important. It could be fixed by deleting "that", although the long dash is jarring. K.W. AHA! The "that" was inserted by MF! :D *LOL* I think that "the theorem states an upper bound" is proper grammatically and conventional mathematically: Other word choices should be considered.  Kiefer.Wolfowitz 04:22, 28 September 2011 (UTC)

  Oops! Malleus Fatuorum 18:30, 29 September 2011 (UTC)

Real vector spaces

• "More generally, any real vector space of (finite) dimension D can be viewed as the set of all possible D-tuples of D real numbers { (v₁, v₂, . . . , v_D) } together with two operations". How can a real vector space be viewed as two operations?

  First, we have the real numbers, which can be considered to a vector space over itself. This means that every pair of real numbers $(r,s)$ can be multiplied $(r,s)\rightarrow r\cdot s$ and added $(r,s)\rightarrow r+s$. More generally, with two dimensions, we can consider the multiplication of a 2-dimensional vector $(r_{1},r_{2})$ by a real number $s$, which forms the scalar-vector product $s\cdot (r_{1},r_{2})=(s\cdot r_{1},s\cdot r_{2})$; every pair of 2-dimensional vectors can be added, thusly $(r_{1},r_{2})+(s_{1},s_{2})=(r_{1}+s_{1},r_{2}+s_{2})$. The operations for higher-dimensional vector-spaces are defined analogously (elementwise).  Kiefer.Wolfowitz 18:39, 30 September 2011 (UTC)

  I understand that a real vector space is a set of real number pairs, but it's the "together with two operations" I'm unhappy about. The operations are specifically addition and multiplication, not any old operations, and they're applied to the vector spaces. Malleus Fatuorum 20:12, 30 September 2011 (UTC)

  Done! "A set on which two operations are defined: Vector addition and scalar-vector multiplication". K.W.

Shapley–Folkman theorem and Starr's corollary

• "Starr used the inner radius to strengthen the conclusion of the Shapley–Folkman theorem". Theorems don't have conclusions.

  The SF theorem is a conditional theorem with an if-then statement: "If the number of sets is greater than the dimension, then ... an inequality is satisfied." K.W.

  Then that probably ought to be explained, because right now it makes no sense to anyone other than a mathematician. Malleus Fatuorum 00:43, 1 October 2011 (UTC)

  That's an excellent point. I'll fix it.  Kiefer.Wolfowitz 01:45, 1 October 2011 (UTC)

• Support I've now read the whole article, and although it's not an easy read, and I'm not a mathematician, I'm persuaded that it's an accurate account that meets the FA criteria. Malleus Fatuorum

Comments

• For the article as a whole, I'm impressed with the meticulous care that has been taken with both the references and with the explanations of mathematical concepts and issues intended at the average reader. I think there were (there always are) some minor issues with "translating math into English" but I think Malleus caught most if not all of that.

Economics

• Other than that I can really only make detailed comments about the "Economics" section. Again, I see no major problems here. There was some slightly awkward wording, as noted above, in the explanation of the graphical derivation of demand from the indifference curves and budget constraints which I reworded somewhat. Someone should probably make sure that in the rewording I didn't solve one problem by creating another.

  I liked your rewording, but I made a few changes. I removed the reduction to relative prices, just because this interesting (projective-geometry, or dual-space) reduction is not needed, and may confuse some readers. I changed one "the optimal basket" to "an optimal basket", as in a non-strictly-convex example.

• I also agree with the comment above that the "supply and demand" graph in the article is not really necessary. The application of the lemma in economics is to a general equilibrium but the graph depicts a partial equilibrium situation. This is fairly minor though.

  I hid the supply-and-demand graph, following your concern and Protonk's. Another editor may well consider restoring it ....  Kiefer.Wolfowitz 15:55, 2 October 2011 (UTC)

• One final thing - though this is more of a matter of taste - in some ways I think it would make more sense to have the "Probability and measure theory" section precede the "Mathematical optimization" section as the first is more general.

  I agree with the abstract/aesthetic value of having the most general topic first, at least for experts. However, abstraction is a menace in popular science-exposition. ;)

  In the case of this article, it would be very bad to begin the applications with probability. That application-section is really a summary-style section that is of greatest use to students who have had at least a course in the "principles" of real analysis (the theory of calculus, like Walter Rudin's "Baby Rudin") and a basic course in probability. Leading with probability would place a road block in the way of many readers.

  The economics application deserves to come first, at least historically. The optimization application(s) could come first, mathematically. However, as discussed above with Protonk, economists have spent a century simplifying consumer theory, and it seems that the indifference-curve approach enables the article to avoid maximizing utility. (If we used utility maximization, for quasiconcave utility functions, then optimization should precede economics, in exposition as well as in abstract-to-specific ordering). K.W.

• Along the side lines, shouldn't Shapley's photo be moved up in the article, perhaps to the section "Starr's 1969 paper and contemporary economics"? I understand that there are aesthetic issues involved as that may make one section over cluttered with images.

  DONE! Thank you for the great suggestions, now and earlier.  Kiefer.Wolfowitz 15:55, 2 October 2011 (UTC)

• Overall, Support.  Volunteer Marek  20:11, 1 October 2011 (UTC)

Summary

All three reviewers have supported the nomination, and the copyscape audit has revealed no problems (and there are none, I vow). I have responded to all actionable complaints and suggestions (IMHO) inside the scope of a FA nomination.

Outside the scope of a FA nomination, imho, there remains two suggestions for expansion of applications. In optimization, Protonk requests an informal explanation of the significance of the Ekeland/Lemarechal result, which should be done in a few days (e.g., following Bertsekas's textbook). Second, in economics, Protonk's request for another example can be done in 2 days, by providing short verbal explanation of an example from Starr's 1969 paper; alas, IMHO, it is impossible to provide a graphical illustration using economic data to illustrate the SF lemma without doing original research.

Finally, I thank my reviewers for their dedication and excellent suggestions, the article's creator David Eppstein for his initiative & brilliant graphics & continued efforts, the peer reviewers Paul Nguyen, Geometry Guy, & TCO, mensch EdJohnson, for great suggestions, Mike Hardy for copyediting and help with formatting, the copy-editing of LK and others, and especially Jakob for his civilized leadership in the Good Article review, which taught me most of my WikiCraft. Thanks again.  Kiefer.Wolfowitz 16:20, 2 October 2011 (UTC) MOVED FROM ABOVE 09:03, 5 October 2011 (UTC)

Keifer, there is way too much bolding in this nomination. Next time, please don't bold so much within your own comments, and please do not repeat reviewer declarations. This actually makes it harder for delegates to see what is going on at a quick glance and makes it more likely that I'll miss something. I've removed some of the bolding here, but not all that should go away.

It's also unnecessary to add a summary because the delegates have to read everything through anyway. If you choose to add a summary anyway, please do it at the bottom. Karanacs (talk) 15:05, 4 October 2011 (UTC)

I had and still have a lot to learn. Best regards,  Kiefer.Wolfowitz 15:42, 4 October 2011 (UTC)

Just to clarify, most of my substantive comments have been dealt with and I feel this article meets our FA standards. Protonk (talk) 19:51, 4 October 2011 (UTC)

Draft

Lede for main page (Hidden, per Ucucha's note, below)  Kiefer.Wolfowitz 08:54, 7 October 2011 (UTC)

In geometry and economics, the Shapley–Folkman lemma describes the Minkowski addition of sets. Lemmas are steps in a mathematical proof of a theorem. Minkowski addition is defined as the addition of the sets' members: for example

{0, 1} + {0, 1} = {0+0, 0+1, 1+0, 1+1} = {0, 1, 2}.

The Shapley–Folkman lemma provides an affirmative answer to the question, "Is the sum of many sets close to being convex?" A set is defined to be convex if every line segment joining two of its points is a subset in the set: For example, the solid disk $\bullet$ is a convex set but the circle $\circ$ is not, because the line segment joining two distinct points $\oslash$ is not a subset of the circle. The Shapley–Folkman lemma suggests that if the number of summed sets exceeds the dimension of the vector space, then their Minkowski sum is approximately convex. The lemma has many applications in economics, where non-convexity is associated with market failures, that is, with inefficient or non-existent economic equilibria. Non-convex sets have been studied by many winners of the Nobel Prize in Economics: Arrow (1972), Aumann (2005), Debreu (1983), Hurwicz (2007), Kantorovich (1975), Koopmans (1975), Krugman (2008), Samuelson (1970), and Solow (1987). (more…)

This draft has 1221 characters, a palindromic number numerologically arguing against further trimming (to reach 1200 characters).

I would suggest that the article appear 11 October 2011, the day after the Nobel Economics Prize is awarded, or perhaps on the day of the Nobel Award Ceremony.  Kiefer.Wolfowitz 09:43, 6 October 2011 (UTC)

• If the article is passed in time you can add this to this TFA/Requests subpage. I don't know if the article will be passed in time, though. The easiest way to get an article passed more quickly is to bring in more reviewers. Some FACs are declined for lack of reviewers even if all the reviews support the article. Protonk (talk) 18:24, 6 October 2011 (UTC)

• Thanks, Protonk!

  I have written neutral announcements at the mathematics and economics projects (and asked for help with the optimization section with the systems and computer-science projects and for help with the measure theory section with the statistics project) and asked (neutrally) for help at individual talk pages, for editors like yourself who helped out at the GA or A-level review.

  I have confidence in this review process, having read Malleu's comments about the FA community and having witnessed SandyGeorgia's good judgment elsewhere many times. The FA reviewers will recognize your and Marek's economic expertise (and perhaps you are known for FA work unbeknown to me) and Malleus's experience and exacting demands with FA articles and the English language. They can also see that mathematicians like David Eppstein, Geometry Guy, Ed Johnston, and Jakob S. have also given the article a careful review.

  I would prefer firstly to allow the FA leadership time and secondly to respect the time-constraints of the math project, before sending out further appeals. I am pleased that you and Marek and Malleus again volunteered to read the article and provide so many thoughtful remarks.

  If October 11th is no longer available, perhaps the article could be scheduled for the Nobel award ceremony week. Otherwise, it can wait a year. Sincerely and with warm regards,  Kiefer.Wolfowitz 20:13, 6 October 2011 (UTC)

  Expertise aside, I think the expectation of multiple reviewers exist so that we don't get tunnel vision and promote a seemingly great article with a serious flaw. It has happened before. I wouldn't send out any more mass appeals (not that doing so is bad) but if you know any really active copywriters now might be the time to drop them a line. Protonk (talk) 21:10, 6 October 2011 (UTC)

  I am confident that the content and prose satisfy the FA guidelines. The article has been stable for about half a year, with only minor changes. The only worry I had was whether the images (especially the animation) satisfy the FA guidelines.  Kiefer.Wolfowitz 21:37, 6 October 2011 (UTC)

  I renewed a few personal requests. At least one reviewer indicated that, given his previous work on the article, that he thought it appropriate to let others decide FA status. (He also cautioned that mathematics articles typically remain open for a while.) He suggested two FA mathematics editors, whom I've now contacted.  Kiefer.Wolfowitz 11:53, 7 October 2011 (UTC)

Please don't draft the TFA blurb here: this page is only for determining whether the article meets the FA criteria. Use the article's talk page (or some other page) instead. Ucucha (talk) 21:48, 6 October 2011 (UTC)

Sorry. I hid the blurb.  Kiefer.Wolfowitz 11:53, 7 October 2011 (UTC)

Thanks. I'd also like to see an image review and a spotcheck of the sources for this article. Ucucha (talk) 13:24, 7 October 2011 (UTC)

I asked for help at the WP page about Images and Media. Is there a description of how to do a spotcheck? Is your "I would like to see ... a spotcheck" an oppose (if this is not done)?

A "spotcheck" is performed when an uninvolved editor checks the source supplied for a given claim in the article to see if the source fully backs up the claim, and to ensure that there are no problems of verbatim copying or close paraphrasing. Usually someone will need to check several different sources throughout the article. This will be difficult in an article with sources as technical as these. – Quadell ^(talk) 19:34, 7 October 2011 (UTC)

Image review: All images are legitimately free, either freely licensed or in the public domain. All necessary information in provided. – Quadell ^(talk) 19:28, 7 October 2011 (UTC)

At the WP:WikiProject Image and Media's Illustration task force, User:FleetCommand complimented the images, which were created by User:David Eppstein for this article (or created by others for other articles).  Kiefer.Wolfowitz 14:16, 8 October 2011 (UTC)

Weak oppose. Let me first add to the praise above for the meticulous work that has gone into this well-referenced and carefully explained article on an important technical topic. The article has much improved since February (where I contributed a partial peer review). Nevertheless, on reading the article closely (having not done so since February), I find that it falls short of the demanding FA criteria in several respects, and so cannot support its promotion at present. Areas where I believe improvements could be made include: clarity of exposition, engaging/brilliant prose, comprehensiveness and organization. I will add detailed remarks shortly, most of which I hope can be easily addressed. Geometry guy 21:41, 7 October 2011 (UTC)

I thank you for your past comments. I shall address whatever suggestions you make to the best of my ability. (I am away from my office until next week, however.)  Kiefer.Wolfowitz 22:08, 7 October 2011 (UTC)

• Leading issues. The first few sentences of the lead already illustrate some of the issues (numbered so that it should not be necessary to interleave replies).
  1. The repetition "In geometry and economics... In mathematics..." is clunky; despite comments made above about readers not knowing what a lemma is, an entire sentence is overkill – why not simply wikilink "Shapley-Folkman Lemma"?
  2. The article does not explain what the Shapley-Folkman-Starr theorem is: it discusses their results, including a Shapley-Folkman theorem and Starr's corollary, but no theorem with that name.
  3. The lead sentence does not define the topic: the Shapley-Folkman(-Starr) results do not describe the Minkowski addition of sets in a vector space (the definition of Minkowski addition does that); they describe the extent to which the Minkowski sum of many sets is approximately convex.
  4. The distinction between "addition" and "sum" is important. "Addition" is a synonym for "summation", the process of adding, not a synonym for "sum", the result of the addition. (We do not say "5 is the addition of 2 and 3".) The lead needs to make this distinction clear for Minkowski addition/sums, so that the terms can be selected and used for maximum clarity in the article.
  5. The lead uses terms such as "summand(-)set" and "sumset" without defining or wikilinking them. A particularly problematic example is "average sumset". I was completely unclear about what this meant until I read section 3.2.
  6. Theorems do have ("hypotheses" and) "conclusions", and I am relaxed about the idea that a theorem may "address" or "concern" a particular question. "The Shapley–Folkman–Starr results suggest..." is a bit too loose for me, however. A naive reviewer might ask "suggest to whom?", but the point of the sentence is to provide an intuitive summary of the results, not make suggestions.
  7. "The Shapley–Folkman–Starr theorem states an upper bound on the distance between the Minkowski sum and its convex hull—the convex hull of the Minkowski sum is the smallest convex set that contains the Minkowski sum." Unnecessary repetition: "its convex hull (the smallest convex set containing it)".
  8. "Their bound on the distance..." Antecedent missing/unclear.
  9. Final paragraph: here and elsewhere, "The Shapley-Folkman do-dah..." is used far too much as the subject of the sentence. Try turning sentences around by looking for other subjects, and cut down on the tiresome "also"s.

"The topic of non-convex sets in economics has been studied by many Nobel laureates..."

I left this to last, as it may be a more substantial issue. This segment of the lead is repeated in the article, but does not really summarize anything. The comprehensiveness criterion really bites here: "it neglects no major facts or details and places the subject in context". The legacy of the Shapley-Folkman Lemma is that results previously confined to convex economics and optimization (relatively easy) could be extended to the non-convex domain (much harder) by averaging (e.g., assuming many agents); this needs to be discussed to place the article in context. The applications section contains some such discussion, but is primarily pedagogical/technical and mixes mathematical, historical and evaluative material. The segment "Starr's 1969 paper and contemporary economics" then ends with a list which cries out for elaboration. Overall the treatment of the economics background, history and legacy for the results falls short of what I would hope for in a featured article. Geometry guy 23:18, 7 October 2011 (UTC)

• Mathematics issues. Considerable effort has been made to explain the mathematics behind the S-F Lemma. This is commendable and appropriate as none of the mathematics is particularly hard (at least in the context of plane geometry), so it ought to be possible to explain it to a wide readership. There are some minor shortcomings in this respect.
  1. Introductory example. (This should not begin "For example": the section title suffices.) I read this having forgotten the statement of the Shapley-Folkman Lemma and found it didn't help me very much. It isn't clearly laid out as an example of Minkowski addition of two or more sets, and quickly goes on to discuss averages, where I had not yet got the point. There are also little distractions such as describing {0,1,2} as a subset of the integers, which seems irrelevant. At some point, the article should probably introduce the averaged Minkowski sum (and motivate why it is a useful notion).
  2. Real vector spaces. Is it necessary to use the language of vector spaces if they are all viewed as Rⁿ? Maybe, but readers could easily be put off. Perhaps it is more natural to use vector spaces, but not completely so: the natural setting for convexity is affine geometry, and later on, the theory makes use of Euclidean distance.
  3. Convex sets... "a non-empty set Q is defined to be convex if, for each pair of its points, every point on the line segment that joins them is a subset of Q". This sentence is incorrect (unless points are viewed as subsets), possibly because of a partial attempt to simplify the exposition. It can be simplified further: "a non-empty set Q is convex if, for each pair of points in Q, every point on the line segment joining them is in Q". In general, it is helpful to unpack the language of subsets, as set membership is easier to discuss in a non-technical way. A small amount of copyediting would be helpful here (and elsewhere).
  4. Convex sets. "Mathematical induction" disrupts the flow and is only need for one implication in the "if and only if". Why not define convex combinations before using them to characterize convexity?
  5. Convex hull... "is the minimal convex set that contains Q. Thus Conv(Q) is the intersection of all the convex sets that cover Q." Here the word "cover" is incorrect: its use implies a collection of sets whose union contains Q. Conv(Q) is the intersection of all convex sets containing Q, and the the fancy word "minimal" can be replaced by "smallest" by the uniqueness (well-definedness) of this set. It may be helpful to introduce the word "convexification" here as a synonym.
  6. Minkowski sum. The "principle of mathematical induction" again. It isn't needed to define the sum of a family, only to show that an iterated binary sum is equal to the sum of the family (and hence is associative).
  7. Convex hulls and Minkowski sums. Yet more induction! Why is it relevant to discuss a snippet of the proof?
  8. Statements. This begins in a somewhat pedestrian fashion with "x in Y and Y=Z implies x in Z". Repetition may be pedagogical, but encyclopedic writing should be concise and to the point.
  9. Statements/lemma. It may be helpful to use a different letter for an element of Conv(Q_n) than for an element of Q_n. This would help to emphasize the point of writing the sum in two parts.
  10. Shapley–Folkman theorem and Starr's corollary. There is a sudden jump in complexity and sophistication here. The article has been holding the reader's hand up to the lemma, but then says "okay, now you're on your own". A short subsection on Euclidean distance, distnace to a subset, circumradius and inner radius would help a lot.
  11. Starr's corollary. Is this really a corollary to the Shapley-Folkman theorem? It provides a sharper estimate. Also, why is "non-convexity" an abuse?
  12. Proofs and computations. "The original proof of the Shapley–Folkman lemma established only the existence of the representation..." What representation? This needs to be clarified. An idea of the proof of the lemma would be nice too.

There are a few math issues in the applications section, but I will discuss that separately. Geometry guy 16:27, 8 October 2011 (UTC)

• Further issues. It remains to discuss the Applications section.
  1. "The Shapley–Folkman lemma enables researchers...and the Shapley–Folkman lemma has renewed research that had been stumped by non-convex sets.'" Why "researchers"? Had "research" really been "stumped"? This informal present/perfect tense approach needs to be backed up or replaced by historical legacy material.
  2. "In all three disciplines, the break-through application of the Shapley–Folkman lemma has been made by a young scientist." This is an editorial observation; such synthesis should be sourced or cut.
  3. "On this set of baskets, an indifference curve is defined for each consumer..." suggests there is only one curve per consumer, whereas in fact the space is foliated by such curves: there is one through each basket.
  4. "An optimal basket of goods occurs where the budget-line supports a consumer's preference set, as shown in the diagram..." There is too much unexplained economics jargon here. What is a budget-line, price vector and endowment vector, and how is the budget line defined in terms of the other quantities? What is an optimal basket? A feasible one? Is the optimal basket really a function? It looks like it could be multi-valued or only partially defined even in the convex case.
  5. I remain uncomfortable with the griffin example. Apart from the question as to whether it is encyclopedic, it is distracting and confusing. What is meant by half a lion or half an eagle? It only makes sense when discussing dead creatures. Why not have a lion for six months of the year and an eagle for the other six? And surely a contemporary zoo keeper would value a griffin much more highly than a lion or an eagle, because of the fortune to be made out of visiting Harry Potter fans. Finally, the footnote gives a perfectly sensible and completely sourced example (an automobile and a boat) so why not use that?
  6. "Previous publications on non-convexity and economics were collected in an annotated bibliography by Kenneth Arrow." This is a slightly odd start to a section. Previous to what? Wouldn't it be better to start with Starr?
  7. "who proved their eponymous lemma and theorem in 'private correspondence' ". No they didn't: they communicated it to Starr in private correspondence or he quoted it thusly.
  8. The mathematical optimization section spends rather a lot of time defining convex functions. The caption to the diagram is more concise.
  9. The opening of the probability and measure theory section is confusing. The essence of the discussion is that if random quantity only takes values in Q, then its average, being a convex combination, must belong to the convex hull of Q.

That completes my review. Geometry guy 17:10, 8 October 2011 (UTC)

Reply to Geometry Guy

I have drafted initial replies in my user space. I agree with most of his comments and I concede merit to the others. In general, I would argue that disagreements remain where I am trying to help first-time readers, with less mathematical background, by repetition.

I acknowledge that GG may well wish that this article was longer and contained more information, but I reply that FA articles need not be perfect; this article contains far more information and has better graphics than any other treatment in world literature. Readers wanting more treatment of e.g. economics should consult the sources given in the article (most of which are missing from even the union of previous articles on this topic), such as Mas-Colell's article on non-convexity and economics (in .pdf format on his home page).

I thank Geometry Guy again for his careful and conscientious scrutiny.  Kiefer.Wolfowitz 10:46, 8 October 2011 (UTC)

You're welcome. My comments are generally provided "as is" for you and the delegates to make what you will of them. However, I see from your initial response that I need to clarify Leading issues#3 (on the lead sentence). The Shapley-Folkman lemma involves two fundamental ingredients of similar importance: Minkowski addition and convexity. The current lead sentence treats these ingredients differently, referring to Minkowski addition and then defining it, while postponing discussion of the role of convexity. This is not defining. It should not be too difficult to work convexity into the lead sentence, e.g., "In geometry and economics, the Shapley–Folkman lemma describes the extent to which the Minkowski sum of a sets in a vector space is approximately convex." I'm sure you can do better than me, but a good lead paragraph should mention both Minkowski addition and convexity in the first sentence, then define them both. Geometry guy 17:39, 8 October 2011 (UTC)

Agreed. The muse was not whispering in my ear, but I'm beginning to hear an improved first sentence.

If you look at my responses in user space (in history), you will notice that I tend to agree with you after first raising some initial (token) resistance.

I can probably return to this Monday-Tuesday.  Kiefer.Wolfowitz 17:59, 8 October 2011 (UTC)

• Oppose. Here are some issues with the article as it stands:
  1. The article never defines "sumset". It begins using the term in the lead, but the non-expert reader may not realize this is meant to be the Minkowski sum.
  2. In the paragraph where the lead discusses the Shapley–Folkman–Starr theorem, the claim is first made that "[t]heir bound on the distance" does not depend on "the number of summand-sets N, when N > D". It goes on to say that "as the number of summands increases to infinity, the bound decreases to zero". My first thought was that these could not both be true, and that the article was in error. It turns out that I was wrong, because I missed the word "average". I think I won't be the only reader to make this error, so I think the paragraph needs to make a bigger contrast between sumsets and average sumsets.
  3. Still in the lead: It's not clear to me what a convexified economy is, not even vaguely. Nor is it clear what kind of equilibrium is meant (a Nash equilibrium, I guess?) or how that's different from a quasi-equilibrium.
  4. In general, the lead spends a lot of time trying to explain the statement of the Shapley–Folkman lemma (and its relatives). That's not the right direction. What the reader needs to learn is, "Why do I care?" Imagine, for example, that I'm a layman who has read a pop economics book, and I know what supply and demand are and what economies are, but I don't know what a convex set or a Minkowski sum is. Why should I learn about the Shapley–Folkman lemma? The lead does not answer this question. The closest it comes is near the end, where it explains that a lot of famous and successful economists have said that the study of non-convex situations is important (the lead has already made it clear that the Shapley–Folkman lemma is important for these). But essentially it's a proof by authority: All these Nobel laureates think it's important, so you should, too! This will not entice a novice reader to continue.
  5. In general, the article is structured like (I almost hate to say this) a math article. First it describes some preliminary notions (real vector spaces, convex sets, Minkowski sums, etc.). Then it states a theorem. Then it states applications. Kind of like "Definition–Theorem–Proof–Corollary". I realize that we all write this way (including me), but we shouldn't, and we especially shouldn't in an encyclopedia article.
  6. Right now, the article lacks a history section. I am going to suggest (this is only a suggestion, and there may be better ways of doing this) that you move some of the material from the Economics subsection of the Applications section into a new "History and Motivation" section immediately following the lead. This could put foundational material (like convexity) into a historical context: Before the Shapley–Folkman lemma, economists studied convex economies. Convexity is ..., and these are economies in which ... and they were important because of ..., but non-convex economies, which are ..., were important because ..., and prior to the Shapley–Folkman lemma nothing was known about non-convex economies. If you combine the foundational material with historical context, you make it more interesting and easier to grasp: It comes with vivid examples of what used to be cutting-edge research. By the time you are done with the historical context, you should have managed to introduce the prerequisite material for the Shapley–Folkman lemma. Then you can state it (and its corollaries and variations). Once that's done, you can move on to other applications.
  7. The optimization section has the same "Definition–Theorem–Proof–Corollary" feel. Again, I think it would be more effective to weave together the history and the prerequisite material.
  8. I was surprised at how short the section on probabilistic applications is. I don't know how important the Shapley–Folkman lemma is in such work, but you mention that it can be used to prove some analogs of standard results for real-valued random variables (like a law of large numbers and a central limit theorem). It would be good to include some more detail about these so that the reader at least knows how the Shapley–Folkman lemma is relevant (you don't necessarily have to state the theorems to prove this).
  9. Also, it might be good to explain in more detail how Lyapunov's theorem is related to the Shapley–Folkman lemma.
• Ozob (talk) 02:33, 8 October 2011 (UTC)

The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.