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Journal of Geometry

, 110:45 | Cite as

Complex Disk Products and Cartesian Ovals

  • Florian BüngerEmail author
  • Siegfried M. Rump
Article
  • 53 Downloads

Abstract

Let \(D_R\), \(D_r\), \(D_S\), \(D_s\) be complex disks with common center 1 and radii RrSs, respectively. We consider the Minkowski products \(A := D_R D_r\) and \(B := D_S D_s\) and give necessary and sufficient conditions for A being a subset or superset of B. Partially, this extends to n-fold disk products \(D_1\ldots D_n\), \(n>2\). It is well-known that the boundaries of A and B are outer loops of Cartesian ovals. Therefore, our results translate to necessary and sufficient conditions under which such loops encircle each other.

Keywords

complex disk products Minkowski products Cartesian ovals 

Mathematics Subject Classification

Primary 53A04 Secondary 14H45 

Notes

References

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    Bünger, F., Rump, S.M.: The determinant of a complex matrix and Gershgorin circles. Electron. J. Linear Algebra 35, 181–186 (2019)MathSciNetCrossRefGoogle Scholar
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    Farouki, R.T., Moon, H.P., Ravani, B.: Minkowski geometric algebra of complex sets. Geometriae Dedicata 85, 283–315 (2001)MathSciNetCrossRefGoogle Scholar
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    Farouki, R.T., Pottmann, H.: Exact Minkowski products of N complex disks. Reliab. Comput. 8, 43–66 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute for Reliable ComputingHamburg University of TechnologyHamburgGermany
  2. 2.Faculty of Science and EngineeringWaseda UniversityTokyoJapan

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