Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

On the volume of tubular neighborhoods of real algebraic varieties


Author: Martin Lotz
Journal: Proc. Amer. Math. Soc. 143 (2015), 1875-1889
MSC (2010): Primary 14P05, 53C65; Secondary 60D05, 15A12
DOI: https://doi.org/10.1090/S0002-9939-2014-12397-5
Published electronically: December 23, 2014
MathSciNet review: 3314098
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of determining the volume of a tubular neighborhood has a long and rich history. Bounds on the volume of neighborhoods of algebraic sets have turned out to play an important role in the probabilistic analysis of condition numbers in numerical analysis. We present a self-contained derivation of bounds on the probability that a random point, chosen uniformly from a ball, lies within a given distance of a real algebraic variety of any codimension. The bounds are given in terms of the degrees of the defining polynomials, and contain as a special case an unpublished result by Ocneanu.


References [Enhancements On Off] (What's this?)

  • [1] Robert J. Adler and Jonathan E. Taylor, Random fields and geometry, Springer Monographs in Mathematics, Springer, New York, 2007. MR 2319516 (2008m:60090)
  • [2] Dennis Amelunxen and Peter Bürgisser, Probabilistic analysis of the Grassmann condition number, Foundations of Computational Mathematics , posted on (November 14, 2013), 1-49., https://doi.org/10.1007/s10208-013-9178-4
  • [3] Dennis Amelunxen, Martin Lotz, Michael B. McCoy, and Joel A. Tropp, Living on the edge: phase transitions in convex programs with random data, Information and Inference , posted on (2014), available at http://imaiai.oxfordjournals.org/content/early/2014/06/28/imaiai.iau005.full.pdf+html., https://doi.org/10.1093/imaiai/iau005
  • [4] Peter Bürgisser, Smoothed analysis of condition numbers, Proceedings of the International Congress of Mathematicians. Volume IV, Hindustan Book Agency, New Delhi, 2010, pp. 2609-2633. MR 2827986 (2012k:65056)
  • [5] Peter Bürgisser, Felipe Cucker, and Martin Lotz, The probability that a slightly perturbed numerical analysis problem is difficult, Math. Comp. 77 (2008), no. 263, 1559-1583. MR 2398780 (2009a:65132), https://doi.org/10.1090/S0025-5718-08-02060-7
  • [6] Peter Bürgisser, Felipe Cucker, and Martin Lotz, Coverage processes on spheres and condition numbers for linear programming, Ann. Probab. 38 (2010), no. 2, 570-604. MR 2642886 (2011d:60036), https://doi.org/10.1214/09-AOP489
  • [7] Shiing-shen Chern, On the kinematic formula in integral geometry, J. Math. Mech. 16 (1966), 101-118. MR 0198406 (33 #6564)
  • [8] James W. Demmel, On condition numbers and the distance to the nearest ill-posed problem, Numer. Math. 51 (1987), no. 3, 251-289. MR 895087 (88i:15014), https://doi.org/10.1007/BF01400115
  • [9] James W. Demmel, The probability that a numerical analysis problem is difficult, Math. Comp. 50 (1988), no. 182, 449-480. MR 929546 (89g:65062), https://doi.org/10.2307/2008617
  • [10] Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207 (92i:53001)
  • [11] Herbert Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418-491. MR 0110078 (22 #961)
  • [12] Alfred Gray, Tubes, 2nd ed., Progress in Mathematics, vol. 221, Birkhäuser Verlag, Basel, 2004. With a preface by Vicente Miquel. MR 2024928 (2004j:53001)
  • [13] Misha Gromov and Larry Guth, Generalizations of the Kolmogorov-Barzdin embedding estimates, Duke Math. J. 161 (2012), no. 13, 2549-2603. MR 2988903, https://doi.org/10.1215/00127094-1812840
  • [14] Ralph Howard, The kinematic formula in Riemannian homogeneous spaces, Mem. Amer. Math. Soc. 106 (1993), no. 509, vi+69. MR 1169230 (94d:53114), https://doi.org/10.1090/memo/0509
  • [15] William Kahan.
    Conserving Confluence Curbs Ill-Condition.
    Computer Science Dept., University of California, Berkeley, 1972.
    Technical Report 6.
  • [16] Daniel A. Klain and Gian-Carlo Rota, Introduction to geometric probability, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1997. MR 1608265 (2001f:52009)
  • [17] Eric J. Kostlan, Statistical Complexity of Numerical Linear Algebra, ProQuest LLC, Ann Arbor, MI, 1985. Thesis (Ph.D.)-University of California, Berkeley. MR 2634377
  • [18] Michael B. McCoy and Joel A. Tropp, From Steiner formulas for cones to concentration of intrinsic volumes, Discrete Comput. Geom. 51 (2014), no. 4, 926-963. MR 3216671, https://doi.org/10.1007/s00454-014-9595-4
  • [19] John W. Milnor, On the Betti numbers of real varieties,
    Proc. Amer. Math. Soc., 15:275-280, 1964.MR 0161339 (28 #4547)
  • [20] John W. Milnor, Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Based on notes by David W. Weaver; Revised reprint of the 1965 original. MR 1487640 (98h:57051)
  • [21] James Renegar, On the efficiency of Newton's method in approximating all zeros of a system of complex polynomials, Math. Oper. Res. 12 (1987), no. 1, 121-148. MR 882846 (88j:65112), https://doi.org/10.1287/moor.12.1.121
  • [22] Luis A. Santaló.
    Integral Geometry and Geometric Probability.
    Cambridge Mathematical Library. Cambridge University Press, 2004.
  • [23] Rolf Schneider and Wolfgang Weil, Stochastic and integral geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. MR 2455326 (2010g:60002)
  • [24] Steve Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 1-36. MR 590817 (83i:65044), https://doi.org/10.1090/S0273-0979-1981-14858-8
  • [25] Jakob Steiner, Ueber parallele Flächen,
    In Karl Weierstrass, editor, Jakob Steiner's gesammelte Werke, pages 173-176. 1881.
  • [26] Hermann Weyl, On the Volume of Tubes, Amer. J. Math. 61 (1939), no. 2, 461-472. MR 1507388, https://doi.org/10.2307/2371513
  • [27] Richard Wongkew, Volumes of tubular neighbourhoods of real algebraic varieties, Pacific J. Math. 159 (1993), no. 1, 177-184. MR 1211391 (94e:14073)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14P05, 53C65, 60D05, 15A12

Retrieve articles in all journals with MSC (2010): 14P05, 53C65, 60D05, 15A12


Additional Information

Martin Lotz
Affiliation: School of Mathematics, The University of Manchester, Alan Turing Building, Oxford Road, Manchester, M139PL, United Kingdom
Email: martin.lotz@manchester.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2014-12397-5
Received by editor(s): April 2, 2013
Received by editor(s) in revised form: September 20, 2013, and September 24, 2013
Published electronically: December 23, 2014
Additional Notes: This research was supported by Leverhulme Trust grant R41617 and a Seggie Brown Fellowship of the University of Edinburgh
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society