Equivariant corner straightening is usually mentioned in the literature without further explanation. What would be a reference where this is done (more or less) carefully for compact Lie group actions on smooth manifolds, or at least for finite group actions?.
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$\begingroup$ What kind of corners are you interested in? The simplest is cubical corners but there is a wide variety of other notions. $\endgroup$ – Ryan Budney Nov 28 '16 at 21:08

$\begingroup$ I phrased the question broadly to gain general understanding of what's possible. For my current applications the corners are as in a disk bundle over a disk. $\endgroup$ – Igor Belegradek Nov 28 '16 at 22:06

1$\begingroup$ There is a somewhat technical but pretty natural argument you can make whenever the corners are convex  meaning the tangent vectors of the manifold that "point in" to the manifold form a convex subspace of the tangent space. This is a class of stratified space that's wider than manifolds with cubical corners. Any kind of sufficiently natural and geometric argument generalizes to the equivariant case. My impression is these arguments tend to require choices that bog the discussion down, and so you mostly see them in masters thesis. . . $\endgroup$ – Ryan Budney Nov 28 '16 at 22:11

$\begingroup$ Any reference would be welcome but I am even more interested in a discussion as to what is possible and what is not. $\endgroup$ – Igor Belegradek Nov 29 '16 at 0:37