On the weighted Kneser-Poulsen conjecture

Károly Bezdek; Robert Connelly

doi:10.1007/s10998-008-8121-4

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Periodica Mathematica Hungarica

Volume/Issue:: Volume 57: Issue 2

On the weighted Kneser-Poulsen conjecture

Authors:

Károly Bezdek

Károly Bezdek University of Calgary Department of Mathematics and Statistics 2500 University drive N.W. Calgary AB Canada T2N 1N4

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and

Robert Connelly

Robert Connelly Cornell University Department of Mathematics Malott Hall, Room 433 Ithaca NY 14853 USA

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Pages:: 121–129

Online Publication Date:: 10 Dec 2008

Publication Date:: 01 Dec 2008

Article Category:: Research Article

DOI:: https://doi.org/10.1007/s10998-008-8121-4

Keywords:: 52A20; 52A37; 52C17; 52C35; flowers of balls; Boolean flower formula; flower weight function; the weighted Kneser-Poulsen conjecture for flowers

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Abstract

Suppose that p = (p1, p2, …, pN) and q = (q1, q2, …, qN) are two configurations in

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{E}^d$$ \end{document}

, which are centers of balls B^d(p_i, r_i) and B^d(q_i, r_i) of radius r_i, for i = 1, …, N. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q, then the volume of the union of the balls does not increase. For d = 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way. Here we show that these generalized Kneser-Poulsen flower conjectures are equivalent to an inequality between certain integrals of functions (called flower weight functions) over

, where the functions in question are constructed from maximum and minimum operations applied to functions each being radially symmetric monotone decreasing and integrable.

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Periodica Mathematica Hungarica

Print ISSN:: 0031-5303

Online ISSN:: 1588-2829

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Periodica Mathematica Hungarica
Language	English
Size	B5
Year of Foundation	1971
Volumes per Year	2
Issues per Year	4
Founder	Bolyai János Matematikai Társulat - János Bolyai Mathematical Society
Founder's Address	H-1055 Budapest, Hungary Falk Miksa u. 12.I/4.
Publisher	Akadémiai Kiadó Springer Nature Switzerland AG
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245. CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	0031-5303 (Print)
ISSN	1588-2829 (Online)