# Best of MathOverflow, or papers inspired by MathOverflow

This is a place to collect MathOverflow success stories!

Was some of your research inspired by something on MathOverflow? Do you know questions & answers that led to interesting research? MathOverflow citations? Open problems solved on MathOverflow? Then add your story in an answer! (One story per answer, please!)

If you want to help get this thread started, you can use this search to find MathOverflow citations on the arXiv or migrate some old stories from tea.mathoverflow.net.

• This question was suggested in this discussion. Aug 12, 2013 at 15:07
• Is it really the case that the old success stories have to be migrated by hand? I'm surprised there's not a way to do this within the stack software or at least write a program to do it. Aug 12, 2013 at 18:22
• @David: Clarify what you're suggesting. What do you propose migrating this way? Aug 12, 2013 at 18:27
• Hi. I felt that if we plan to have a single unified place for success stories (e.g. if someone wants to come and write an article about MO) then it doesn't make sense to have one place at tea for pre-migration and one here for post-migration. So the old success stories should be here, since it appears we can't add new ones to that old page. But it would take a long time to move them over one by one. I wish it could be done all at once. Aug 12, 2013 at 19:01
• @David: The thread at tea is mostly other stuff so it's best to do it manually. Aug 12, 2013 at 19:10
• @DavidWhite tangentially, but one could add things in the old thread.
– user9072
Aug 12, 2013 at 21:15
• Many papers were inspired by questions raised on MO, but has any credit been given to those who raised the questions in the publications? I am just wondering.
– qed
Aug 24, 2013 at 16:09
• @qed I know for a fact that in many instances, perhaps most, the MO question-asker is given credit and citations. Further, I know of several cases, and I expect that there are many more, in which the question-asker was invited to join as co-author. Dec 26, 2016 at 15:02
• I wonder if the title should be changed from "Best of MathOverflow" to "Publications resulting from MathOverflow" (or maybe "inspired by"?). It has in fact developed into publication citing. Apr 9, 2017 at 1:57
• @JosephO'Rourke What you write is perhaps true. OTOH the ones which MO community considers the best are quite likely to rise to the top based on voting, so in this sense it might still be a fitting title. I'll point out that Todd Trimble recently commented on the title of this thread: Some people might feel uneasy citing their own work at a thread entitled "Best of MathOverflow", but perhaps that title should be interpreted broadly Jan 21, 2018 at 9:49
• This is sort of a duplicate of this closed question: mathoverflow.net/q/11846/1345 Nov 19, 2019 at 18:41
• @Ian I think the idea was that meta is a more appropriate place for this question than main was. Dec 3, 2019 at 22:37
• As the link to tea given at the end of the post does not work now, I'll add a link to Wayback Machine at least here in a comment. Jun 14 at 5:31

## 57 Answers

Mark Sapir solved an interesting open problem in group theory in this answer to a question of Narutaka OZAWA.

A meta-answer: http://search.arxiv.org:8081/?query=mathoverflow&in= returns a list of 197 papers on the arXiv which mention MathOverflow.

Nearly all of these are actual citations, with a small number of papers about MathOverflow itself, and some number of papers which mention MathOverflow without giving full attribution according to the guidelines.

Perhaps at some point it would be interesting to analyze this full collection (which is clearly a significant superset of the things mentioned in other answers here).

• This is incomplete. The query does not return arxiv.org/abs/1302.1659v2, which mentions MO. Sep 12, 2013 at 9:12
• The issue there seems to be that MO is only mentioned in a URL, so "mathoverflow" is not found as a word by the search. There are probably others which insert a space, as in "Math Overflow" and are similarly missed, but I can't get arXiv search to work for me right now to check. Sep 13, 2013 at 11:08
• It is great to have your query, but we should recognize that not probably not every citation of MathOverflow rises to the level of a "success story". At least some of the papers arising in your query, for example, including some of my own papers there, are just making a small citation about a minor matter. So perhaps we should interpret the question as asking for instances of work that is more deeply connected with or inspired by something on MathOverflow. Sep 13, 2013 at 19:16
• @JoelDavidHamkins, agreed, perhaps I should have mentioned this search URL somewhere else. Sep 14, 2013 at 3:22
• Oh, I think it is fine here (and a similar query was mentioned already in the question). Sep 14, 2013 at 3:24
• Hmmm, this is weird - four year laters, that list has increased to 201, which seems to me to be a much lower increment than you'd expect. So maybe that search is missing some important bits. Nov 29, 2017 at 15:20

In April 2013 I discussed with Alan Sokal the following conjecture: if $$P$$ is a real polynomial with the property $$|P(z)| then some power of $$P$$ has positive coefficients. We did not prove it at that time.

In August, Ofer Zeitouni asked on MO to describe all possible limits of the so-called empirical measures of polynomials with positive coefficients. He needed this for his research on random polynomials. It immediately crossed my mind, that a proof of the Sokal's conjecture stated above will imply the answer to Zeitouni's question.

The final result was a paper by Walter Bergweiler, Alan Sokal and myself, https://www.math.purdue.edu/~eremenko/newprep.html where we prove the a necessary and sufficient conditions on a real polynomial for some power of it to have positive coefficients, and give an answer to Zeitouni's question. Zeros of polynomials with real positive coefficients

Edit. This story has a continuation, also related to MO. Another related question was asked on MO, and one of the answers contained a reference to a published proof of what I called "Sokal conjecture". It was essentially the same as our proof. So the net outcome was only a paper answering Zeitouni's question.

• In turn, that answer (and the methods in the paper) will be used by Subhro Ghosh and I in a forthcoming work, which we hope to post soon. Sep 28, 2013 at 19:50

Our article D. Brumleve, J. D. Hamkins, and P. Schlicht, “The mate-in-n problem of infinite chess is decidable,” LNCS 7318(2012):78-88 was inspired directly by Richard Stanley's question Decidability of chess on an infinite board.

Abstract. Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-n problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most n moves. The main theorem of this article, confirming a conjecture of the second author and C. D. A. Evans, establishes that the mate-in-n problem of infinite chess is computably decidable, uniformly in the position and in n. The proof proceeds by showing that the mate-in-n problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable. An alternative argument proceeds via Presburger arithmetic, which is capable of interpreting the mate-in-n problem of infinite chess.

(This was a collaboration truly born on MathOverflow, as some of the authors have never met in person...)

• Is Richard Stanley also a collaborator to the article? Jun 19, 2015 at 5:06
• No; we cite his question, but he is not a co-author or collaborator. Jun 19, 2015 at 10:12
• And by that, your definition of "collaborator" refers to? Jun 26, 2015 at 23:17
• I had meant the ordinary meaning of "collaborator." In mathematical research, this would be someone with whom you work together on a research project, making a joint intellectual effort. Here is a list of my own research collaborators: jdh.hamkins.org/collaborators Jun 27, 2015 at 0:13
• So is it basically just a place where you thank people for helping? Is a "collaborator" an objective thing or is it totally subjective and you can list anyone you wanted that had helped you tangentially? What would be the bare minimum to qualify for a "collaborator"? Jul 2, 2015 at 12:05
• I would refer you to a dictionary or to english.stackexchange.com for advice on use of the word "collaborator." Jul 2, 2015 at 21:29
• I actually mean.. is it a specific terminology used in maths research? Jul 8, 2015 at 10:41

This is an old and self-indulgent story; but it was such a charmingly unexpected bonus from my early use of MathOverflow, that I think it deserves to be recorded somewhere (my apologies for its length!):

tl;dr: As a serendipitous consequence of this MathOverflow question, the second answerer invited me to give my first-ever seminar talk as a grad student.

At the time I was a 2nd year grad student in topology, working on a project related to knot signatures. Specifically, I was hoping to relate this paper by Kirk and Livingston to some possibly novel computations I'd made, but I was having some basic difficulties, leading to this MathOverflow question. My question didn't mention the paper, since my confusion was quite preliminary to its content. In addition to a great answer by Emerton, I got another great answer from a mysterious user "Paul." In an illuminating response to my follow-up comment, Paul even mentioned the paper I'd been reading! In my surprised reply, I explained that this paper was in fact directly responsible for my question.

Paul eventually revealed by email that in a miraculous coincidence, he was in fact P. Kirk, one of the authors of that original, motivating paper! (This possibility had certainly never occurred to me). After more email exchanges, and more due to his kindness than my results, Paul actually invited me to talk in the Bloomington topology seminar, to discuss things in person. This led to my first ever "invited" seminar talk, and a truly fantastic visit to Bloomington, very formative as well as informative!

My computations themselves were never actually published, but they did make it into my thesis, which is on the arXiv (Chapter 3). This story, however, is not in my thesis! I'm glad I could record it somewhere. Please edit if appropriate!

• Why not link to your thesis on the arXiv? Jun 30, 2015 at 7:33
• good point, done! for some reason, when I wrote the post, it felt too self-indulgent to promote my thesis as well ;) Jun 30, 2015 at 10:02
• This is a great success story, and also points out one of the advantages of using real names on MO. When I can identify a poster to be a motivated student I am much more likely to go out of my way to be helpful than if the OP is anonymous. Aug 18, 2015 at 17:32

Ben Green's paper on (not) computing the Möbius function arose from this question on MathOverflow.

Abstract. Any function $$F : \{1,\dots,N\} \rightarrow \{-1,1\}$$ such that $$F(x)$$ can be computed from the binary digits of $$x$$ using a bounded depth circuit is orthogonal to the Möbius function $$\mu$$ in the sense that $$\frac{1}{N} \sum_{x \leq N} \mu(x)F(x) = o_{N \rightarrow \infty}(1)$$. The proof combines a result of Linial, Mansour and Nisan with techniques of Kátai and Harman-Kátai, used in their work on finding primes with specified digits.

The MO question

Norms of Commutators

is referenced in a paper by Ozawa, Schechtman and me that will appear in the PNAS. arXiv:1202.0986

Tom Church, Melody Chan, and Joshua Grochow just posted their paper "Rotor-routing and spanning trees on planar graphs" to the arXiv here. It answers this MO question which was asked by Jordan Ellenberg.

Our article, C. D. A. Evans and J. D. Hamkins, Transfinite game values in infinite chess, where we investigate the range of transfinite game values arising in infinite chess, grew directly out of Johan Wästlund's question Checkmate in $\omega$ moves?. In particular, we define the omega one of chess $\omega_1^{\frak{Ch}}$ to be the supremum of the ordinal game values that arise in the positions of infinite chess.

Abstract. In this article, we investigate the transfinite game values arising in infinite chess, providing both upper and lower bounds on the supremum of these values — the omega one of chess — denoted by $\omega_1^{\mathfrak{Ch}}$ in the context of finite positions and by $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$ in the context of all positions, including those with infinitely many pieces. For lower bounds, we present specific positions with transfinite game values of $\omega$, $\omega^2$, $\omega^2\cdot k$ and $\omega^3$. By embedding trees into chess, we show that there is a computable infinite chess position that is a win for white if the players are required to play according to a deterministic computable strategy, but which is a draw without that restriction. Finally, we prove that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infinite three-dimensional chess is as large as it can be, namely, true $\omega_1$.

The article is 38 pages, with 18 figures detailing many interesting positions of infinite chess. My co-author Cory Evans holds the chess title of U.S. National Master.

Follow the links to see the chess positions, such as the following, which has value $\omega^2\cdot 4$.

The paper A Counterexample to a Conjecture of Schwartz by Brandt, Chudnovsky, Kim, Liu, Norin, Scott, Seymour, and Thomassé answers this MO question of Felix Brandt. The question asks whether a weakened form of Schwartz’ Conjecture (a popular conjecture in Social Choice Theory) is true. The paper proves that even this weakened form of the conjecture is false, thus resolving Schwartz’ Conjecture in the negative. I think this is a nice example where one area calls on the expertise of another, and the call is answered.

Abstract. In 1990, motivated by applications in the social sciences, Thomas Schwartz made a conjecture about tournaments which would have had numerous attractive consequences. In particular, it implied that there is no tournament with a partition $A, B$ of its vertex set, such that every transitive subset of $A$ is in the out-neighbour set of some vertex in $B$, and vice versa. But in fact there is such a tournament, as we show in this paper, and so Schwartz’ conjecture is false. Our proof is non-constructive and uses the probabilistic method.

• This is nice, but I am not sure how it qualifies: Brandt posted the question, so he was already thinking about it, and then Brandt and his collaborators solved it (perhaps because no other answer was suggested here). Oct 9, 2013 at 21:56
• As I understand it, the question was completely unknown to all the non-Brandt authors before the MO question. I attended a talk by Ilhee Kim where he said that he saw the question on MO and started working on it. He then told the problem to Paul Seymour (his PhD advisor), and then eventually many other people got involved and solved it. So without MO, the problem would likely still be open. Oct 10, 2013 at 10:14
• Ah, great! I did not see them mentioning this in the paper, that would have been nice. Thank you! Oct 10, 2013 at 13:07

This paper (details below) by Zhen Lin Low and Aaron Mazel-Gee cites not just MO but:

This collaboration would not have happened without the ‘Homotopy Theory’ chat room on MathOverflow.

arXiv.org > math > arXiv:1409.8192

From fractions to complete Segal spaces

Zhen Lin Low, Aaron Mazel-Gee

We show that the Rezk classification diagram of a relative category admitting a homotopical version of the two-sided calculus of fractions is a Segal space up to Reedy-fibrant replacement. In particular, the Rezk classification diagram of a closed model category in the sense of Quillen is a complete Segal space up to Reedy-fibrant replacement, resolving a conjecture of Rezk.

This MO question was asked in December of 2011, in line with a reference request for a senior thesis on odd perfect numbers completed in 1978. Subsequently, the OP has tried numerous ways to get hold of the thesis's author.

On August 24, 2013 Jim (Condict) Grace (the thesis's author) popped in to MO to respond to the original question.

• I don't see how this qualifies as an answer to the question. It's not research inspired by something on MathOverflow; it's not a citation of MathOverflow; it's not an open problem solved on MathOverflow. Aug 26, 2013 at 0:23
• It's a success story, isn't it? Sep 23, 2013 at 15:11

My MO question Conjugation of group extensions was answered by YCor. As a result, we wrote a joint note Conjugate complex homogeneous spaces with non-isomorphic fundamental groups published in C. R. Acad. Sci. Paris, Ser. I, 353 (2015) 1001–1005.

• Congratulations! I think that this information should be added to the original MO question as well.
– j.c.
Mar 5, 2016 at 19:45

My joint article with Justin Palumbo, The rigid relation principle, a new weak choice principle (Mathematical Logic Quarterly 58(6):394-398, 2012) grew out of our answers to my question, Does every set admit a rigid binary relation? (and how is this related to the Axiom of Choice?), which grew out of Mike Shulman's question, A rigid type of structure that can be put on every set?, on which I had made my very first post upon coming here to MathOverflow.

Abstract. The rigid relation principle asserts that every set admits a rigid binary relation. This follows from the axiom of choice, because well-orders are rigid, but we prove that it is neither equivalent to the axiom of choice nor provable in Zermelo-Fraenkel set theory without the axiom of choice. Thus, it is a new weak choice principle. Nevertheless, the restriction of the principle to sets of reals (among other general instances) is provable without the axiom of choice.

I am frequently asked whether exponentials are "linearly independent". That is if we have a sequence of distinct complex numbers $\lambda_j$, whether $$\sum_{j} a_j\exp(\lambda_j z)\equiv 0$$ implies that all $a_j=0$. If the linear dependence above holds in all complex plane, this question was answered by A. F. Leontjev. All his books exist only in Russian, but I was asked this question so frequently that I wrote a short note, Linear independence of exponentials, explaining Leontjev's results, and posted it on Internet.

Then a question was asked on MO, On linear independence of exponentials, what if the linear dependence relation holds only for real $z$ (the $\lambda_j$ and $a_j$ are still complex). I could not find an answer in Leontjev's papers, and the result was a preprint http://www.math.purdue.edu/~eremenko/dvi/exp2.pdf

The question whether there is a non surjective bounded linear operator on $$\ell_\infty$$ that has dense range was answered in this paper by Amir Bahman Nasseri, Gideon Schechtman, Tomasz Tkocz, and me. An interesting aspect of the proof is that it uses a theorem proved by computer scientists to get a counterexample. So, in some sense, this question about operators on a non separable Banach space is connected to computer science!

The analog of the famous law of iterated logarithm for maximum eigenvalue of a random Gaussian matrix was asked here. Zeitouni's MO-answer was expanded (after significant effort) to a full answer for the limsup (including constants) and a partial answer for the liminf by Elliot Paquette and Ofer Zeitouni arxiv.org/abs/1505.05627 !

As acknowledged in my note Explicit additive decomposition of norms on $\mathbb{R}^2$, it was sparked by answers by Noam D. Elkies and Suvrit Sra on MathOverflow Absolute value inequality for complex numbers and Bill Johnson’s comments there. Part of the note also used the answer by Bill Johnson to a related question posted by me.

• Now published in Amer. Math. Monthly, May 2016, vol.123, no.5 pp.491-496. Jun 4, 2016 at 0:43

A nice question by Michael Hardy, How many rearrangements must fail to alter the value of a sum before you conclude that none do?, led to a recent 6-author collaboration, 5 or 6 of whom are MO patrons if I'm not mistaken.

• A. Blass, J. Brendle, W. Brian, J.D. Hamkins, M. Hardy, and P.B. Larson, The rearrangement number (manuscript under review).

See also this answer by Joel David Hamkins, https://mathoverflow.net/a/214779/2926, for more information. Please update this answer, Joel David, when this is published in a journal!

• I receive some AMS email alerts and just noticed some familiar MO names (otherwise I probably wouldn't have opened the article). It looks like the paper is now available in Transaction of the AMS at doi.org/10.1090/tran/7881 and references the original question (and the question links back to this meta post). Oct 7, 2019 at 16:20
1. In my paper Invariant curves and semiconjugacies of rational functions [Fund. Math. 219 (2012), no. 3, 263–270; MR3001243; DOI:10.4064/fm219-3-5], I proved a theorem characterizing Jordan analytic invariant curves of rational functions or certain type. My theorem implies that all such curves must be algebraic, but there was no examples except circles. I asked on Overflow whether there are any other examples, there was no answer for some time, then I offered a bounty.

The required examples were constructed by Peter Mueller. By that time my paper was already published, and I could not mention these examples in it, but Peter promised to include them in his own paper Decompositions of rational functions over real and complex numbers and a question about invariant curves.

Circles and rational functions

1. When I asked this question "Analytic function avoiding elements of the modular group", I was working on a problem about Painleve VI. The answer was very illuminating, and eventually led to a solution of my problem on Painleve VI, which resulted in this paper, where I acknowledge the MO discussion.
• I will add at least in a comment eudml link and arXiv link - simply because the paper (or the preprint) is accessible here without needing any type of subscription. Jan 21, 2018 at 10:59
• Thanks. I added the link. Jan 21, 2018 at 13:02

The paper Quartic graphs with every edge in a triangle by Florian Pfender and Gordon Royle grew out of the MO question 4-regular graphs with every edge in a triangle.

Keith Kearnes, together with co-authors Emil Kiss and Ágnes Szendrei, recently published a solution to Varieties where every algebra is free in this arxiv preprint. They prove a result under an even weaker hypothesis: that "a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine vector spaces over a division ring".

• Minor clarification: it later turned out that the title question had been answered by Steven Givant in his 1975 thesis. So the novelty in the the arxiv preprint is the weakening of the hypothesis, which was suggested as an afterthought in the MO question.
– Tim Campion Mod
Sep 2, 2017 at 20:31

This MO question was the starting point for a joint work with Tao Mei where we study radial multipliers on the von Neumann algebras of hyperbolic groups. The paper is entitled Complete boundedness of the Heat Semigroups on the von Neumann Algebra of hyperbolic groups, and as the title suggests it contains among other a proof that, to our surprise, the heat semigroup, although not positive, is bounded on the von Neumann algebras of hyperbolic groups. The arXiv version is here, and it will soon appear in Transactions of the AMS.

According to Christian Stump, his paper "On a New Collection of Words in the Catalan Family" (Journal of Integer Sequences, vol. 17 (2014), article 14.7.1) is a long version of his answers to a MathOverflow question asked by Vince Vatter and to a follow-up question asked by David Speyer.

The question, "How hard is reconstructing a permutation from its differences sequence?" posed by Mohammad Al-Turkistany, was answered by Marzio De Biasi, who then wrote a paper, "Permutation Reconstruction from Differences," published in the Electronic Journal of Combinatorics (2014):

We prove that the problem of reconstructing a permutation $\pi_1,\ldots,\pi_n$ of the integers $[1\ldots n]$ given the absolute differences $|\pi_{i+1}-\pi_i|$, $i=1,\ldots,n−1$, is 𝖭𝖯-complete.

Not sure if my recent paper "Equivalence: an attempt at a history of the idea" qualifies as one of the "best of Mathoverflow or papers inspired by Mathoverflow". But I am sure Mathoverflow was a force to keep me motivated for a journey that started 13 years ago into the long and rich history of equivalence.

It was 4 years and 6 months ago that I asked on MO: "Who introduced the terms “equivalence relation” and “equivalence class”?" When I asked the question I was kind of full of myself to know nearly every corner of the relevant history, and the question was kind of let me find even that bit that I don't know. But, suddenly, there it was @Francois Ziegler's answer and then his comment to my own answer. Wow, it was much more than I asked. Basically, it opened up my eyes to something in front of me all the times, but I had failed to see it all the times. That answer was a new beginning for something that had started 6 years and a half year ago and continued for another 4 years and a half year!
The paper has been dedicated to David Fowler for the reasons mentioned in the paper and here (link to meta-MO post). But, I believe both Fowler and I should thank Francois for his short enlightening answer. Here is the abstract of the paper, hoping it deserves the name of David Fowler, Christopher Zeeman, Jeremy Gray, and Francois Ziegler who directly or indirectly, knowingly or unknowingly, encouraged me to finish my journey.

This paper proposes a reading of the history of equivalence in mathematics. The paper has two main parts. The first part focuses on a relatively short historical period when the notion of equivalence is about to be decontextualized, but yet, has no commonly agreed-upon name. The method for this part is rather straightforward: following the clues left by the others for the ‘first’ modern use of equivalence. The second part focuses on a relatively long historical period when equivalence is experienced in context. The method for this part is to strip the ideas from their set-theoretic formulations and methodically examine the variations in the ways equivalence appears in some prominent historical texts. The paper reveals several critical differences in the conceptions of equivalence at different points in history that are at variance with the standard account of the mathematical notion of equivalence encompassing the concepts of equivalence relation and equivalence class.

Hannah Cairns's proof of Perron's theorem (discussed in this MathOverflow question) has been published in The American Mathematical Monthly as Perron’s Theorem in an Hour.

This paper,

Roman Karasev, Jan Kynčl, Pavel Paták, Zuzana Safernová, and Martin Tancer. "Bounds for Pach's selection theorem and for the minimum solid angle in a simplex." arXiv:1403.8147 (2014). Discrete & Computational Geometry. 54:610-636 (2015).

cites my answer to Boris Bukh's question, Angle of a regular simplex, in the discussion of their theorem giving an upper bound on the minimum solid angle of a $d$-simplex.

Incidentally, they pose a very nice question:

Is it true that the minimum solid angle of a $d$-simplex is at most the solid angle of a regular $d$-simplex?

The answer is Yes for $d \le 4$.

An unpublished open problem posed by Adam Chalcraft, Does every polyomino tile $$\mathbb R^n$$ for some $$n$$?, received considerable attention when I posted it here on MO. (Of all the questions that I have posed on MO that aren't soft questions, it has received the most upvotes—69 as of this writing.) It was solved by Vytautas Gruslys, Imre Leader, and Ta Sheng Tan, who learned about it from MO, in

Vytautas, Gruslys, Imre Leader, Ta Sheng Tan, Tiling with arbitrary tiles, Proc. Lond. Math. Soc. (3) 112 No. 6 (2016) 1019–1039, doi:10.1112/plms/pdw017, arXiv:1505.03697

• And just to close the loop: They provided a positive answer. Dec 5, 2019 at 0:26

Mohammad Ghomi answered the question Shortest closed curve to inspect a sphere, in a paper, Shortest closed curve to inspect a sphere, posted to the arXiv (https://arxiv.org/abs/2010.15204), whose PDF is available here, verifying the conjecture that the shortest curve is four consecutive semicircles each of length $$\pi$$ for a unit sphere.