On universal covers in normed planes
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Abstract
In 1914, Lebesgue posed his famous universal cover problem for the Euclidean plane which is still unsolved. We present the first explicit investigation of this problem for arbitrary normed planes. A convex body K in a finite dimensional real Banach space is called universal cover if any set of diameter 1 can be covered by a congruent copy of K; if ”congruent copy“ is replaced by ”translate”, then K is called strong universal cover. We describe the smallest regular hexagons, the smallest equilateral triangles, and the smallest squares (all these figures defined in the sense of the norm) that are strong universal covers in normed planes. In addition, the paper contains a new characterization of Radon planes. In view of universal covers having ball shape we shortly discuss also Jung constants of real Banach spaces, and a known result on Borsuk’s partition problem is reproved.
© 2013 by Walter de Gruyter GmbH & Co.
Articles in the same Issue
- Masthead
- A class of lattices and boolean functions related to the Manickam–Miklös–Singhi conjecture
- Blocking semiovals containing conics
- On universal covers in normed planes
- Stability results for some classical convexity operations
- Pseudo-embeddings and pseudo-hyperplanes
- Asymptotic estimates on the time derivative of entropy on a Riemannian manifold
- Metrics in the family of star bodies
- Ellipsoid characterization theorems
- Logarithmic limit sets of real semi-algebraic sets
Articles in the same Issue
- Masthead
- A class of lattices and boolean functions related to the Manickam–Miklös–Singhi conjecture
- Blocking semiovals containing conics
- On universal covers in normed planes
- Stability results for some classical convexity operations
- Pseudo-embeddings and pseudo-hyperplanes
- Asymptotic estimates on the time derivative of entropy on a Riemannian manifold
- Metrics in the family of star bodies
- Ellipsoid characterization theorems
- Logarithmic limit sets of real semi-algebraic sets
