Circle configurations in strictly convex normed planes
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Margarita Spirova
Abstract
We present extensions of results of P. J. Kelly [Amer. Math. Monthly 57: 677–678, 1950] on the so-called re-entrant property of circles in strictly convex normed (or Minkowski) planes, and also further properties of circles, well known for the Euclidean plane, are generalized for all strictly convex Minkowski planes. More precisely, we present “Minkowskian analogues” of the philosophical symbol Yin-Yang, of the Arbelos, a special case of the famous Apollonius problem on circles touching each other, and one of the Sangaku-circles problems (coming from the Japanese Temple Geometry). The latter is remarkable since the consideration of this type of problems in the Euclidean plane requires the use of inversion or of the Pythagorean Theorem (i.e., of tools having no analogues in normed planes). Finally we observe that a strictly convex normed plane which, in addition, is smooth can be considered as a flat Möbius plane where, for example, Apollonius' problem is solved.
© de Gruyter 2010
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- A combinatorial approach to Alexander–Hirschowitz's Theorem based on toric degenerations
- Triply periodic minimal surfaces which converge to the Hoffman–Wohlgemuth example
- Total absolute horospherical curvature of submanifolds in hyperbolic space
- The case of equality for an inverse Santaló functional inequality
- Circle configurations in strictly convex normed planes
- Rank four vector bundles without theta divisor over a curve of genus two
- Affine Tallini Sets and Grassmannians
- Geometry of self-dual flats over a PID on a polarity
- Logarithmic comparison theorem versus Gauss–Manin system for isolated singularities
- A kinematic formula for the total absolute curvature of intersections
- Reducible Veronese surfaces
- Effective non-vanishing conjectures for projective threefolds
Articles in the same Issue
- A combinatorial approach to Alexander–Hirschowitz's Theorem based on toric degenerations
- Triply periodic minimal surfaces which converge to the Hoffman–Wohlgemuth example
- Total absolute horospherical curvature of submanifolds in hyperbolic space
- The case of equality for an inverse Santaló functional inequality
- Circle configurations in strictly convex normed planes
- Rank four vector bundles without theta divisor over a curve of genus two
- Affine Tallini Sets and Grassmannians
- Geometry of self-dual flats over a PID on a polarity
- Logarithmic comparison theorem versus Gauss–Manin system for isolated singularities
- A kinematic formula for the total absolute curvature of intersections
- Reducible Veronese surfaces
- Effective non-vanishing conjectures for projective threefolds
