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Dedicated to our friend Imre Bárány on the occasion of his 50-th birthday. Abstract For any λ > 1 we construct a periodic and locally finite packing of the plane with ellipses whose λ-enlargement covers the whole plane. This answers a question of Imre Bárány. On the other hand, we show that if C is a packing in the plane with circular discs of radius at… (More)

This paper gives a partial confirmation of a conjecture of P. that the total curvature of a shortest path on the boundary of a convex polyhedron in R 3 cannot be arbitrarily large. It is shown here that the conjecture holds for a class of polytopes for which the ratio of the radii of the circumscribed and inscribed ball is bounded. On the other hand, an… (More)

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