Kneser–Poulsen conjecture

Importance: High ✭✭✭
Subject: Geometry
Keywords: pushing disks
Recomm. for undergrads: no
Posted by: tchow
on: September 25th, 2008

\begin{conjecture} If a finite set of unit balls in $\mathbb{R}^n$ is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease. \end{conjecture}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/} This problem dates from the mid-1950's. The planar case was solved by Bezdek and Connelly in 2003, who also showed that the area of the intersection does not increase, and that the result holds even if the disks have unequal radii. In higher dimensions the problem remains open.

The conjecture is known to hold if the rearrangement can be executed by a continuous motion such that the distance between every pair of centers monotonically increases throughout the motion.

Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

*[BC] K. Bezdek and R. Connelly, \href[Pushing disks apart: The Kneser-Poulsen conjecture in the plane]{http://arxiv.org/abs/math.MG/0108098}, J. Reine Angew. Math. 553 (2002), 221--236.

*[K] M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass, Arch. Math. 6 (1955), 382--390.

*[P] E. T. Poulsen, Problem 10, Math. Scand. 2 (1954), 346.


* indicates original appearance(s) of problem.