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Equiangular Subspaces in Euclidean Spaces


A set of lines through the origin is called equiangular if every pair of lines defines the same angle, and the maximum size of an equiangular set of lines in \(\mathbb {R}^n\) was studied extensively for the last 70 years. In this paper, we study analogous questions for k-dimensional subspaces. We discuss natural ways of defining the angle between k-dimensional subspaces and correspondingly study the maximum size of an equiangular set of k-dimensional subspaces in \(\mathbb {R}^n\). Our bounds extend and improve a result of Blokhuis.

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  1. 1.

    Asimov, D.: The grand tour—a tool for viewing multidimensional data. SIAM J. Sci. Stat. Comput. 6(1), 128–143 (1985)

  2. 2.

    Balla, I., Dräxler, F., Keevash, P., Sudakov, B.: Equiangular lines and spherical codes in Euclidean space. Invent. Math. 211(1), 179–212 (2018)

  3. 3.

    Barg, A., Yu, W.-H.: New bounds for equiangular lines. In: Barg, A., Musin, O.R. (eds.) Discrete Geometry and Algebraic Combinatorics. Contemporary Mathematics, vol. 625, pp. 111–121. American Mathematical Society, Providence (2014)

  4. 4.

    Blokhuis, A.: Polynomials in finite geometries and combinatorics. In: Walker, K. (ed.) Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 187, pp. 35–52. Cambridge University Press, New York (1993)

  5. 5.

    de Caen, D.: Large equiangular sets of lines in Euclidean space. Electron. J. Combin. 7, Art. No. 55 (2000)

  6. 6.

    Calderbank, A.R., Hardin, R.H., Rains, E.M., Shor, P.W., Sloane, N.J.A.: A group-theoretic framework for the construction of packings in Grassmannian spaces. J. Algebr. Comb. 9(2), 129–140 (1999)

  7. 7.

    Conway, J.H., Hardin, R.H., Sloane, N.J.A.: Packing lines, planes, etc.: packings in Grassmannian spaces. Exp. Math. 5(2), 139–159 (1996)

  8. 8.

    Delsarte, P., Goethals, J.M., Seidel, J.J.: Bounds for systems of lines, and Jacobi polynomials. Philips Res. Rep. 30, 91–105 (1975)

  9. 9.

    Deutsch, F.: The angle between subspaces of a Hilbert space. In: Singh, S.P. (ed.) Approximation Theory, Wavelets and Applications. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 454, pp. 107–130. Kluwer, Dordrecht (1995)

  10. 10.

    Dhillon, I.S., Heath Jr., R.W., Strohmer, T., Tropp, J.A.: Constructing packings in Grassmannian manifolds via alternating projection. Exp. Math. 17(1), 9–35 (2008)

  11. 11.

    Dixmier, J.: Étude sur les variétés et les opérateurs de Julia, avec quelques applications. Bull. Soc. Math. Fr. 77, 11–101 (1949)

  12. 12.

    Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998)

  13. 13.

    Friedrichs, K.: On certain inequalities and characteristic value problems for analytic functions and for functions of two variables. Trans. Am. Math. Soc. 41(3), 321–364 (1937)

  14. 14.

    Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics. Springer, New York (2001)

  15. 15.

    Krein, M.G., Krasnoselskii, M.A., Milman, D.P.: On the defect numbers of linear operators in Banach spaces and on some geometric questions. Trudy Inst. Mat. Akad. Nauk Ukrain. SSR 11, 97–112 (1948) (in Russian)

  16. 16.

    Lemmens, P.W.H., Seidel, J.J.: Equiangular lines. J. Algebra 24, 494–512 (1973)

  17. 17.

    Lemmens, P.W.H., Seidel, J.J.: Equi-isoclinic subspaces of Euclidean spaces. Indag. Math. 35, 98–107 (1973)

  18. 18.

    van Lint, J.H., Seidel, J.J.: Equilateral point sets in elliptic geometry. Indag. Math. 28, 335–348 (1966)

  19. 19.

    Wong, Y.: Differential geometry of Grassmann manifolds. Proc. Natl. Acad. Sci. USA 57(3), 589–594 (1967)

  20. 20.

    Yokonuma, T.: Tensor Spaces and Exterior Algebra. Translations of Mathematical Monographs. American Mathematical Society, Providence (1992)

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Author information

Correspondence to Benny Sudakov.

Additional information

Research supported in part by SNSF Grant 200021-175573.

Editor in Charge: János Pach

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Balla, I., Sudakov, B. Equiangular Subspaces in Euclidean Spaces. Discrete Comput Geom 61, 81–90 (2019).

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  • Equiangular lines
  • Subspaces
  • Grassmannian
  • Principal angles
  • Polynomial method

Mathematics Subject Classification

  • 52C35
  • 05D99