Polyhedral combinatorics of bisectors
-
Aryaman Jal
and
Katharina Jochemko
Abstract
For any polyhedral norm, the bisector of two points is a polyhedral complex. We study combinatorial aspects of this complex. We investigate the sensitivity of the presence of labelled maximal cells in the bisector relative to the position of the two points. We thereby extend work of Criado, Joswig and Santos (2022) who showed that for the tropical distance function the presence of maximal cells is encoded by a polyhedral fan, the bisection fan. We initiate the study of bisection cones and bisection fans with respect to arbitrary polyhedral norms. In particular, we show that the bisection fan always exists for polyhedral norms in two dimensions. Furthermore, we determine the bisection fan of the ℓ1-norm and the ℓ∞-norm as well as the discrete Wasserstein distance in arbitrary dimensions. Intricate combinatorial structures, such as the resonance arrangement, make their appearance. We apply our results to obtain bounds on the combinatorial complexity of the bisectors.
Funding statement: AJ and KJ were partially supported by the Wallenberg AI, Autonomous Systems and Software Program funded by the Knut and Alice Wallenberg Foundation. KJ was moreover supported by a Hanna Neumann Fellowship of the Berlin Mathematics Research Center Math+, by grant 2018-03968 of the Swedish Research Council and by the Göran Gustafsson Foundation.
Acknowledgements
The authors are grateful to the anonymous reviewer for careful reading of the manuscript and helpful comments. The authors would like to thank Michael Joswig, Benjamin Schröter and Francisco Criado for helpful and inspiring discussions. In particular, they are grateful to Michael Joswig for hosting them at TU Berlin during spring 2022 when parts of this work were undertaken. They are also very thankful to Benjamin Schröter for his invaluable help with polymake.
Communicated by: F. Santos
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Articles in the same Issue
- Frontmatter
- Polyhedral combinatorics of bisectors
- Explicit construction of decomposable Jacobians
- Some rigidity results on q-solitons
- Quasimorphisms on nonorientable surface diffeomorphism groups
- On nearly optimal paper Moebius bands
- Desargues’ Theorem in Laguerre Planes
- Simple obstructions and cone reduction
- The Bogomolov–Gieseker–Koseki inequality on surfaces with canonical singularities in arbitrary characteristic
- The simultaneous lattice packing-covering constant of octahedra
- An Abel–Jacobi theorem for metrized complexes of Riemann surfaces
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Articles in the same Issue
- Frontmatter
- Polyhedral combinatorics of bisectors
- Explicit construction of decomposable Jacobians
- Some rigidity results on q-solitons
- Quasimorphisms on nonorientable surface diffeomorphism groups
- On nearly optimal paper Moebius bands
- Desargues’ Theorem in Laguerre Planes
- Simple obstructions and cone reduction
- The Bogomolov–Gieseker–Koseki inequality on surfaces with canonical singularities in arbitrary characteristic
- The simultaneous lattice packing-covering constant of octahedra
- An Abel–Jacobi theorem for metrized complexes of Riemann surfaces
- Exploring the interplay of semistable vector bundles and their restrictions on reducible curves
- Another proof of Segre’s theorem about ovals
