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A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n.

Is there a citeable reference for a proof of this result? For the sake of being definite, let's say that “citeable” means indexed by Mathematical Reviews or Zentralblatt, or available on arXiv.

The answer https://mathoverflow.net/a/4516 gives two references for this theorem, neither of which is citeable in the above sense: online notes and an obscure book, impossible to locate. There are many citeable sources that state this result without a proof.

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    $\begingroup$ Why doesn't someone write a short note, stick it on the arXiv? And then get it published in one of American Mathematical Monthly, Expositiones Mathematicae (Elsevier, though!), Confluentes Mathematici etc (sourced from mathoverflow.net/questions/15366/…). Also, some relevant discussion/references is in ncatlab.org/nlab/show/ball $\endgroup$
    – David Roberts ♦
    Commented Mar 2, 2015 at 22:39
  • $\begingroup$ @DavidRoberts: Good suggestion, although the note might not be as short as one might expect it to be: the proof in online notes occupies 3 pages. $\endgroup$
    – Dmitri Pavlov
    Commented Mar 2, 2015 at 23:09
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    $\begingroup$ @DavidRoberts: Actually I was discussing the existence of differentiable good open covers with Urs Schreiber when he visited Göttingen, and the result of the discussion was that one doesn't need any curvature or convexity radius estimates. Here's the proposed proof: to construct a differentiable good open cover of a smooth manifold use partitions of unity to construct a Riemannian metric and take the set of all geodesically convex subsets. They are closed under intersection and the inverse of the geodesic flow transforms any geodesically convex open set into a star-shaped open set. QED $\endgroup$
    – Dmitri Pavlov
    Commented Mar 3, 2015 at 0:16
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    $\begingroup$ @GregFriedman have a look at this more recent answer (mathoverflow.net/a/212595/11211) to the MO question linked by the OP with a link to a manuscript copy done by Erwann Aubry. In any case, I managed to buy a used copy of the book "Calcul Différentiel" by Gonnord and Tosel, and the proof there is indeed much neater than the one in Dirk Ferus's lecture notes. Unfortunately, they provide no references for it, which is a bit strange (for they do so for other results in the book) and also begs the question of whether this proof is actually due to the authors themselves or not... $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Jun 11, 2016 at 3:17
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    $\begingroup$ @DavidRoberts the theorem of existence of geodesically convex neighborhoods is due to J.H.C. Whitehead (Convex Regions in the Geometry of Paths, Quart. J. Math. 3 (1932) 33-42). A proof valid for any manifold with an affine connection (not just Riemannian) may be found in the charming (although unfortunately out-of-print) little book of Noel J. Hicks (the same from the Cartan-Ambrose-Hicks theorem), Notes on Differential Geometry (Van Nostrand, 1965), Section 9.4, pp. 134-136. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Jun 11, 2016 at 3:32

2 Answers 2

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The result in the following paper implies that open star-shaped domainin $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$. But, in your case, a diffeomorphism can be obtained along the same lines.

K. W. Kwun. “Uniqueness of the open cone neighborhood”. Proc. Amer. Math. Soc. 15 (1964), pp. 476–479.

Postscript. Instead of a reference one could write the following lines:

Any star-shaped open set $\Omega$ is a union of a nested sequence of star-shaped open regions $\Omega_0\subset \Omega_1\subset\dots$ such that $\partial\Omega_n$ is a graph of a smooth Lipschitz function in the polar coordinates. We can assume that $\Omega_0$ is a disc; in particular $\Omega_0$ is diffeomorphic to $\mathbb{R}^n$. Observe that for each $n$ there is a diffeomorphism $\phi_n\colon\bar \Omega_{n-1}\to\bar\Omega_n$; moreover we can assume that $\phi_n$ fix all points away from a tiny neightborhood of $\partial\Omega_n$. In particular, it can be arranged that for any $x_0\in \Omega_0$, the sequence defined by $x_n=\phi_n(x_{n-1})$ stabilizes after finitely many steps. Define $f(x_0)=x_n$ for all large $n$, and observe that $f\colon \Omega_0\to\Omega$ is a diffeomorphism.

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    $\begingroup$ Kwun's proof heavily uses gluing of piecewise continuous maps along closed subsets. Such constructions cannot be adapted to the setting of diffeomorphisms without considerable modifications, e.g., using smooth bump functions. While this probably can be done, it would require one to write a proof that is longer than the one presented in mathoverflow.net/questions/4468/…, which defeats the very purpose of citing a result instead of writing it up. $\endgroup$
    – Dmitri Pavlov
    Commented Nov 18, 2022 at 17:35
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    $\begingroup$ However, for $n\neq 4$ homeomorphism implies diffeomorphism :) $\endgroup$
    – Marco Golla
    Commented Nov 19, 2022 at 12:23
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This result is stated as Exercise 7 in Chapter 8 of Bröcker and Jänich's book Introduction to Differential Topology (p. 86 of the English translation). This may or may not count as a citeable reference.

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  • $\begingroup$ (It's certainly citeable. But one might hope for a published proof rather than just a brief indication of how a proof might go.) $\endgroup$
    – Dan Ramras
    Commented Nov 18, 2022 at 19:30
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    $\begingroup$ I meant a citeable reference with a proof, since there are many known citeable sources that state the result without a proof, e.g., the Bott–Tu book states it, and several others. $\endgroup$
    – Dmitri Pavlov
    Commented Nov 18, 2022 at 21:51
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    $\begingroup$ Fair enough. My recollection, from graduate school, was that someone pointed me to this reference saying that it was the closest thing to a proof in the literature. $\endgroup$
    – Dan Ramras
    Commented Nov 19, 2022 at 4:15

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