Journal of Symplectic Geometry

Volume 17 (2019)

Number 6

Bang’s problem and symplectic invariants

Pages: 1579 – 1611

DOI: https://dx.doi.org/10.4310/JSG.2019.v17.n6.a1

Authors

Arseniy Akopyan (Institute of Science and Technology, Klosterneuburg, Austria; and Institute for Information Transmission Problems RAS, Moscow, Russia)

Roman Karasev (Department of Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Russia; and Institute for Information Transmission Problems RAS, Moscow, Russia)

Fedor Petrov (Steklov Mathematical Institute, Saint-Petersburg, Russia)

Abstract

We consider the Tarski–Bang problem about covering of convex bodies by planks. The results of this kind give a lower bound on the sum of widths of planks (regions between a pair of parallel hyperplanes) covering a given convex body.

Previously we have applied some notions of symplectic geometry to study convex bodies, and here we show that the symplectic techniques may be useful in this problem as well. We are able to handle some particular cases with the symplectic techniques, and show that the general cases would follow from a certain “subadditivity conjecture” in symplectic geometry, motivated by the results of K. Ball. We also prove several related results by more elementary methods.

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A.A. is supported by People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA Grant Agreement No. [291734]. R.K. is supported by the Federal professorship program grant 1.456.2016/1.4 and the Russian Foundation for Basic Research grants 18-01-00036 and 19-01-00169. F.P. is supported by the Russian Foundation for Basic Research grant 17-01-00433.

Received 22 March 2016

Accepted 3 August 2018

Published 17 January 2020