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Some open problems

The problems here were chosen because they can be stated in a simple
and elementary way. I don't go into references to results concerning
them, but if you intend to think about them seriously I encourage you
to check the literature !
The angel and the devil.

Unfolding polyhedra: given a convex polyhedron (in Euclidean
3-space), is it possible to cut it along some of its edges so that the
resulting surface is connected and can be unfolded on a plane, without
self-intersection ?

The Kneser-Poulsen conjecture: given $N$ balls in Euclidean
$n$-space, suppose one changes their position so that the distance
between the centers of any two balls is greater in the new
configuration. Is the volume of the intersection of the balls then
smaller than in the first configuration ? Is the volume of the union
of the balls greater ?

Is there a triangulation of the torus with all vertices of degree
6 except one of degree 5 and one of degree 7 ? I heard this problem
from Ken Stephenson, who said it was open.

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