Abstract
We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the d-dimensional Euclidean space. When the radius r of the balls is large, this volume can be approximated by a polynomial of r, which will be computed up to an \(O(r^{d-3})\) error term. We study how the top coefficients of this polynomial depend on the set of the centers. It is known that in the case of the union of the balls, the top coefficients are some constant multiples of the intrinsic volumes of the convex hull of the centers. Thus, the coefficients in the general case lead to generalizations of the intrinsic volumes, in particular, to a generalization of the mean width of a set. Some known results on the mean width, along with the theorem on its monotonicity under contractions are extended to the “Boolean analogues” of the mean width.
Dedicated to Károly Bezdek and Egon Schulte on the occasion of their 60th birthdays.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
K. Bezdek, Private communication. Retrospective Workshop on Discrete Geometry, Optimization, and Symmetry, The Fields Institute, Toronto, November 25–29, 2013
K. Bezdek, R. Connelly, Pushing disks apart–the Kneser-Poulsen conjecture in the plane. J. Reine Angew. Math. 553, 221–236 (2002). https://doi.org/10.1515/crll.2002.101
V. Capoyleas, J. Pach, On the perimeter of a point set in the plane, in Discrete and computational geometry (New Brunswick, NJ, 1989/1990), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 6 (Amer. Math. Soc., Providence, RI, 1991), pp. 67–76
B. Csikós, On the volume of flowers in space forms. Geom. Dedicata 86(1–3), 59–79 (2001). https://doi.org/10.1023/A:1011983123985
B. Csikós, A Schläfli-type formula for polytopes with curved faces and its application to the Kneser-Poulsen conjecture. Monatsh. Math. 147(4), 273–292 (2006). https://doi.org/10.1007/s00605-005-0363-7
S. Givant, P. Halmos, Introduction to Boolean algebras, in Undergraduate Texts in Mathematics (Springer, New York, 2009). https://doi.org/10.1007/978-0-387-68436-9
I. Gorbovickis, Strict Kneser-Poulsen conjecture for large radii. Geom. Dedicata 162, 95–107 (2013). https://doi.org/10.1007/s10711-012-9718-0
Y. Gordon, M. Meyer, On the volume of unions and intersections of balls in Euclidean space, in Geometric aspects of functional analysis (Israel, 1992–1994), Oper. Theory Adv. Appl., vol. 77 (Birkhäuser, Basel, 1995), pp. 91–101
M. Gromov, Monotonicity of the volume of intersection of balls, in Geometrical aspects of functional analysis (1985/86), Lecture Notes in Math., vol. 1267 (Springer, Berlin, 1987), pp. 1–4. https://doi.org/10.1007/BFb0078131
M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass. Arch. Math. (Basel) 6, 382–390 (1955)
E.T. Poulsen, Problem 10. Math. Scandinavica 2, 346 (1954)
R. Schneider, Convex bodies: the Brunn-Minkowski theory, in Encyclopedia of Mathematics and its Applications, vol. 151, expanded edn. (Cambridge University Press, Cambridge, 2014)
Acknowledgements
This research was supported by the Hungarian National Science and Research Foundation OTKA K 112703. Part of the research was done in the academic year 2014/15, while the author enjoyed the hospitality of the MTA Alfréd Rényi Institute of Mathematics as a guest researcher.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Csikós, B. (2018). On the Volume of Boolean Expressions of Large Congruent Balls. In: Conder, M., Deza, A., Weiss, A. (eds) Discrete Geometry and Symmetry. GSC 2015. Springer Proceedings in Mathematics & Statistics, vol 234. Springer, Cham. https://doi.org/10.1007/978-3-319-78434-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-78434-2_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-78433-5
Online ISBN: 978-3-319-78434-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)