Geometric Aspects of Functional Analysis pp 137-149 | Cite as

# A Remark on Projections of the Rotated Cube to Complex Lines

## Abstract

Motivated by relations with a symplectic invariant known as the “cylindrical symplectic capacity”, in this note we study the expectation of the area of a minimal projection to a complex line for a randomly rotated cube.

## Notes

### Acknowledgements

The authors would like to thank the anonymous referee for helpful comments and remarks, and in particular for his/her suggestion to elaborate more on the symplectic topology background which partially served as a motivation for the current note. The second-named author was partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, starting grant No. 637386, and by the ISF grant No. 1274/14.

## References

- 1.S. Artstein-Avidan, V.D. Milman, Logarithmic reduction of the level of randomness in some probabilistic geometric constructions. J. Funct. Anal.
**235**(1), 297–329 (2006)MathSciNetCrossRefMATHGoogle Scholar - 2.S. Artstein-Avidan, A. Giannopoulos, V.D. Milman,
*Asymptotic Geometric Analysis, Part I.*Mathematical Surveys and Monographs, vol. 202 (American Mathematical Society, Providence, RI, 2015)Google Scholar - 3.T. Cieliebak, H. Hofer, J. Latschev, F. Schlenk, Quantitative symplectic geometry, in
*Dynamics, Ergodic Theory, and Geometry*, vol. 54. Mathematical Sciences Research Institute Publications (Cambridge University Press, Cambridge, 2007), pp. 1–44Google Scholar - 4.T. Figiel, J. Lindenstrauss, V.D. Milman, The dimension of almost spherical sections of convex bodies. Bull. Am. Math. Soc.
**82**(4), 575–578 (1976); expanded in “The dimension of almost spherical sections of convex bodies”. Acta Math.**139**, 53–94 (1977)Google Scholar - 5.A.Y. Garnaev, E.D. Gluskin, The widths of a Euclidean ball. Dokl. Akad. Nauk SSSR
**277**(5), 1048–1052 (1984)MathSciNetMATHGoogle Scholar - 6.G. Giannopoulos, V.D. Milman, A. Tsolomitis, Asymptotic formula for the diameter of sections of symmetric convex bodies. JFA
**233**(1), 86–108 (2005)MathSciNetCrossRefMATHGoogle Scholar - 7.E.D. Gluskin, Norms of random matrices and diameters of finite-dimensional sets. Mat. Sb. (N.S.)
**120**(162) (2), 180–189, 286 (1983)Google Scholar - 8.E.D. Gluskin, Y. Ostrover, Asymptotic equivalence of symplectic capacities. Comm. Math. Helv.
**91**(1), 131–144 (2016)MathSciNetCrossRefMATHGoogle Scholar - 9.M. Gromov, Pseudoholomorphic curves in symplectic manifolds. Invent. Math.
**82**, 307–347 (1985)MathSciNetCrossRefMATHGoogle Scholar - 10.H. Hofer, E. Zehnder,
*Symplectic Invariants and Hamiltonian Dynamics*(Birkhäuser Advanced Texts, Birkhäuser Verlag, 1994)CrossRefMATHGoogle Scholar - 11.B.S. Kašin, The widths of certain finite-dimensional sets and classes of smooth functions (Russian). Izv. Akad. Nauk SSSR Ser. Mat.
**41**(2), 334–351, 478 (1977)Google Scholar - 12.Y. Makovoz, A simple proof of an inequality in the theory of n-widths, in
*Constructive Theory of Functions (Varna, 1987)*(Publ. House Bulgar. Acad. Sci., Sofia, 1988), pp. 305–308Google Scholar - 13.D. McDuff, D. Salamon,
*Introduction to Symplectic Topology*. Oxford Mathematical Monographs (Oxford University Press, New York, 1995)MATHGoogle Scholar - 14.V.D. Milman, A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies. Funk. Anal. i Prilozhen.
**5**(4), 28–37 (1971)Google Scholar - 15.V.D. Milman, Spectrum of a position of a convex body and linear duality relations, in
*Israel Mathematical Conference Proceedings (IMCP)*vol. 3. Festschrift in Honor of Professor I. Piatetski-Shapiro (Part II) (Weizmann Science Press of Israel, 1990), pp. 151–162Google Scholar - 16.V.D. Milman, G. Schechtman,
*Asymptotic Theory of Finite-Dimensional Normed Spaces*. Lecture Notes in Mathematics, vol. 1200 (Springer, Berlin, 1986)Google Scholar - 17.B.S. Mitjagin, Random matrices and subspaces, in
*Geometry of Linear Spaces and Operator Theory*(Russian) (Yaroslav. Gos. Univ., Yaroslavl, 1977), pp. 175–202Google Scholar - 18.Y. Ostrover, When symplectic topology meets Banach space geometry, in
*Proceedings of the ICM*, Seoul, vol. II (2014), pp 959–981.Google Scholar - 19.A. Pajor, N. Tomczak-Jaegermann, Subspaces of small codimension of finite-dimensional Banach spaces. Proc. Am. Math. Soc.
**97**(4), 637–642 (1986)MathSciNetCrossRefMATHGoogle Scholar - 20.G. Pisier,
*The Volume of Convex Bodies and Banach Space Geometry.*Cambridge Tracts in Mathematics, vol. 94 (Cambridge University Press, Cambridge, 1989)Google Scholar - 21.C. Schütt, Entropy numbers of diagonal operators between symmetric Banach spaces. J. Approx. Theory
**40**(2), 121–128 (1984)MathSciNetCrossRefGoogle Scholar - 22.S. Szarek, On Kashin’s almost Euclidean orthogonal decomposition of
*l*_{n}^{1}. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**26**(8), 691–694 (1978)MathSciNetMATHGoogle Scholar