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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 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.)

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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 Mar 2 '15 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 Mar 2 '15 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 Mar 3 '15 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 Jun 11 '16 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 Jun 11 '16 at 3:32

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