Advertisement

On the Number of Flats Tangent to Convex Hypersurfaces in Random Position

  • Khazhgali Kozhasov
  • Antonio LerarioEmail author
Article
  • 15 Downloads

Abstract

Motivated by questions in real enumerative geometry (Borcea et al., in Discrete Comput Geom 35(2):287–300, 2006; Bürgisser and Lerario, in J Reine Angew Math,  https://doi.org/10.1515/crelle-2018-0009, 2018; Megyesi and Sottile, in Discrete Comput Geom 33(4):617–644, 2005; Megyesi et al., in Discrete Comput Geom 30(4):543–571, 2003; Sottile and Theobald, in Trans Am Math Soc 354(12):4815–4829, 2002; Proc Am Math Soc 133(10):2835–2844, 2005; in: Goodman et al., in Surveys on discrete and computational geometry. AMS, Providence, 2008) we investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a probabilistic point of view (here by “convex hypersurfaces” we mean that these hypersurfaces are boundaries of convex sets). More precisely, we say that smooth convex hypersurfaces \(X_1, \ldots , X_{d_{k,n}}\subset {\mathbb {R}}\text {P}^n\), where \(d_{k,n}=(k+1)(n-k)\), are in random position if each one of them is randomly translated by elements \(g_1, \ldots , g_{{d_{k,n}}}\) sampled independently from the orthogonal group with the uniform distribution. Denoting by \(\tau _k(X_1, \ldots , X_{d_{k,n}})\) the average number of k-dimensional projective subspaces (k-flats) which are simultaneously tangent to all the hypersurfaces we prove that
$$\begin{aligned} \tau _k(X_1, \ldots , X_{d_{k,n}})={\delta }_{k,n} \cdot \prod _{i=1}^{d_{k,n}}\frac{|\Omega _k(X_i)|}{|\text {Sch}(k,n)|}, \end{aligned}$$
where \({\delta }_{k,n}\) is the expected degree from [6] (the average number of k-flats incident to \(d_{k,n}\) many random \((n-k-1)\)-flats), \(|\text {Sch}(k,n)|\) is the volume of the Special Schubert variety of k-flats meeting a fixed \((n-k-1)\)-flat (computed in [6]) and \(|\Omega _k(X)|\) is the volume of the manifold \(\Omega _k(X)\subset \mathbb {G}(k,n)\) of all k-flats tangent to X. We give a formula for the evaluation of \(|\Omega _k(X)|\) in terms of some curvature integral of the embedding \(X\hookrightarrow {\mathbb {R}}\text {P}^n\) and we relate it with the classical notion of intrinsic volumes of a convex set:
$$\begin{aligned} \frac{|\Omega _{k}(\partial C)|}{|\text {Sch}(k, n)|}=4V_{n-k-1}(C),\quad k=0, \ldots , n-1. \end{aligned}$$
As a consequence we prove the universal upper bound:
$$\begin{aligned} \tau _k(X_1, \ldots , X_{d_{k,n}})\le {\delta }_{k, n}\cdot 4^{d_{k,n}}. \end{aligned}$$
Since the right hand side of this upper bound does not depend on the specific choice of the convex hypersurfaces, this is especially interesting because already in the case \(k=1, n=3\) for every \(m>0\) we can provide examples of smooth convex hypersurfaces \(X_1, \ldots , X_4\) such that the intersection \(\Omega _1(X_1)\cap \cdots \cap \Omega _1(X_4)\subset \mathbb {G}(1,3)\) is transverse and consists of at least m lines. Finally, we present analogous results for semialgebraic hypersurfaces (not necessarily convex) satisfying some nondegeneracy assumptions.

Keywords

Enumerative geometry Real Grassmannians Schubert calculus Integral geometry 

Mathematics Subject Classification

14N15 14Pxx 52A22 60D05 

Notes

Acknowledgements

We wish to thank P. Bürgisser, K. Kohn, F. Sottile and T. Theobald for interesting discussions and the anonymous referees for their useful comments.

References

  1. 1.
    Amelunxen, D.: Geometric Analysis of the Condition of the Convex Feasibility Problem. PhD Thesis, Universität Paderborn (2011)Google Scholar
  2. 2.
    Amelunxen, D., Bürgisser, P.: Probabilistic analysis of the Grassmann condition number. Found. Comput. Math. 15(1), 3–51 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Basu, S., Lerario, A., Lundberg, E., Peterson, C.: Random fields and the enumerative geometry of lines on real and complex hypersurfaces (2016). arXiv:1610.01205
  4. 4.
    Borcea, C., Goaoc, X., Lazard, S., Petitjean, S.: Common tangents to spheres in \({\mathbb{R}}^3\). Discrete Comput. Geom. 35(2), 287–300 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brönnimann, H., Devillers, O., Dujmovic, V., Everett, H., Glisse, M., Goaoc, X., Lazard, S., Na, H.-S., Whitesides, S.: On the number of lines tangent to four convex polyhedra. In: 14th Canadian Conference on Computational Geometry (CCCG’02), pp. 113–117, Lethbridge, Canada (2002)Google Scholar
  6. 6.
    Bürgisser, P., Lerario, A.: Probabilistic Schubert calculus. J. Reine Angew. Math. (2018).  https://doi.org/10.1515/crelle-2018-0009
  7. 7.
    Gao, F., Hug, D., Schneider, R.: Intrinsic volumes and polar sets in spherical space. Math. Notae 41, 159–176 (2003) (2001/2002) (in Spanish)Google Scholar
  8. 8.
    Howard, R.: The Kinematic Formula in Riemannian Homogeneous Spaces. Memoirs of the American Mathematical Society, vol. 106(509). American Mathematical Society, Providence (1993)Google Scholar
  9. 9.
    Kohn, K., Nødland, B.I.U., Tripoli, P.: Secants, bitangents, and their congruences. In: Smith, G.G., Sturmfels, B. (eds.) Combinatorial Algebraic Geometry. Fields Institute Communications, vol. 80, pp. 87–112. Fields Institute for Research in Mathematical Sciences, Toronto (2017)CrossRefGoogle Scholar
  10. 10.
    Kozlov, S.E.: Geometry of real Grassmannian manifolds. I, II, III. J. Math. Sci. (N.Y.) 100(3), 2239–2253, 2254–2268 (2000)Google Scholar
  11. 11.
    Megyesi, G., Sottile, F.: The envelope of lines meeting a fixed line and tangent to two spheres. Discrete Comput. Geom. 33(4), 617–644 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Megyesi, G., Sottile, F., Theobald, T.: Common transversals and tangents to two lines and two quadrics in \({\mathbb{P}}^3\). Discrete Comput. Geom. 30(4), 543–571 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sottile, F., Theobald, T.: Lines tangent to \(2n-2\) spheres in \({\mathbb{R}}^n\). Trans. Am. Math. Soc. 354(12), 4815–4829 (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Sottile, F., Theobald, T.: Real \(k\)-flats tangent to quadrics in \({\mathbb{R}}^n\). Proc. Am. Math. Soc. 133(10), 2835–2844 (2005)CrossRefzbMATHGoogle Scholar
  15. 15.
    Sottile, F., Theobald, T.: Line problems in nonlinear computational geometry. In: Goodman, J.E., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry. Contemporary Mathematics, vol. 453, pp. 411–432. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.SISSATriesteItaly

Personalised recommendations