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Assume we have a smooth bounded domain $\Omega \subset \mathbb{R}^d$, and two points at the boundary $x, y$. I want to show that there exists an integer $n$ such that one can draw an affine path joining $x$ to $y$ in $n$ steps. By an affine path I mean a path which picks up a direction at every contact with the boundary, follows this direction until it hits the boundary again, and pick a new direction at this rebound, and so on...

So in a strictly convex set, the number $n$ is 1 for any couple of points such that $x \ne y$, two for the couple $(x,x)$ (starting from $x$, we can rebound anywhere but at $x$ in one step and come back to $x$ for the second rebound). In general, starting from $x$, we can choose the direction, draw the corresponding segment inside the domain until a point $z \in \partial \Omega$, then choose another direction... and so on until we hit $y$. In general a result like the density of the set of orbits starting at a point will be a sufficient condition (but will also be way more than what we need, since at every rebound we can go from one orbit to the next one), but I am unsure that it holds without further conditions.

Also note that by connectedness, a path already exist, so it seems quite natural that an affine path should exist as well. I have tried to twist this path to obtain something but could not get what I was looking for.

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You can think of pathological geometries such as the domain of $(x,y)$ defined by

$$0.9x^2\sin\frac\pi x\le y\le1.1x^2\sin\frac\pi x, x\in[0,1].$$

A path from $(0,0)$ to $(1,0)$ must make an infinity of bends, though the curve is everywhere differentiable.

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  • $\begingroup$ Great example. However it seems to me that this curve is not even $C^2$, and thus that this domain is not smooth (by which I mean it is a $C^{\infty}$-manifold with boundary). Am I missing something ? $\endgroup$ – Gâteau-Gallois Oct 11 '18 at 8:48
  • $\begingroup$ @Gâteau-Gallois: can we show that no oscillatory curve with an accumulation of inflection points can't be smooth ? $\endgroup$ – Yves Daoust Oct 11 '18 at 8:57
  • $\begingroup$ I believe this is exactly the property I'm looking for. Thank you, as it gives me a new way to look at the problem. But I'm not familiar enough with the definition of inflection points, and of the minimum regularity that would be require for this property to hold. Do you know of any reference where I can find such material ? $\endgroup$ – Gâteau-Gallois Oct 11 '18 at 11:24
  • $\begingroup$ @Gâteau-Gallois: absolutely not, sorry. $\endgroup$ – Yves Daoust Oct 11 '18 at 12:21

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