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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
Published electronically: December 23, 2014
MathSciNet review: 3314098
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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 The bounds are given in terms of the degrees of the defining polynomials, and contain as a special case an unpublished result by Ocneanu.

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Additional Information

Martin Lotz
Affiliation: School of Mathematics, The University of Manchester, Alan Turing Building, Oxford Road, Manchester, M139PL, United Kingdom

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

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