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Bisecting measures with hyperplane arrangements

Part of: Discrete geometry Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  31 October 2019

ALFREDO HUBARD  and
ROMAN KARASEV
ALFREDO HUBARD
Affiliation:
Université Paris-Est Marne-la-Vallée, LIGM. UMR 8049, CNRS, ENPC, ESIEE, UPEM, F- 77454, Marne-la-Vallée, France. e-mail : [email protected]
ROMAN KARASEV
Affiliation:
Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700. Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia 127994. e-mail : [email protected]
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Abstract

We show that provided that n is a power of two, any nD measures in ℝn can be bisected by an arrangement of D hyperplanes.

MSC classification

Primary: 52C35: Arrangements of points, flats, hyperplanes 55M20: Fixed points and coincidences 55M35: Finite groups of transformations (including Smith theory)
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 3 , November 2020 , pp. 639 - 647
Copyright
© Cambridge Philosophical Society 2020

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References

REFERENCES

Bárány, I. , Hubard, A., and Jerónimo, J.. Slicing convex sets and measures by a hyperplane. Discrete and Computational Geometry 39(1) (2008), 67–75.CrossRefGoogle Scholar
Barba, L. and Schnider, P.. Sharing a pizza: bisecting masses with two cuts. In Proceedings of the 29th Canadian Conference on Computational Geometry, CCCG’17 (July 2017).Google Scholar
Guillemin, V. and Pollack, A.. Differential Topology (2010).CrossRefGoogle Scholar
Matoušek, J.. Using the Borsuk–Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (Springer Publishing Company, Incorporated, 2007).Google Scholar
Tao, T.. An Introduction to Measure Theory. Graduate Studies in Math. (American Mathematical Society, 2013).Google Scholar
Thom, R.. Quélqués propriétés globales des variétés différentiables. Comment. Math. Helv. 28 (1954), 17–86.CrossRefGoogle Scholar
Varadhan, S.R.S.. Probability Theory (American Mathematical Society, 2001).Google Scholar