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Balanced partitions of vector sequences

✍ Scribed by Imre Bárány; Benjamin Doerr

Publisher
Elsevier Science
Year
2006
Tongue
English
Weight
108 KB
Volume
414
Category
Article
ISSN
0024-3795

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✦ Synopsis

Let d, r ∈ N and • be any norm on R d . Let B denote the unit ball with respect to this norm. We show that any sequence v 1 , v 2 , . . . of vectors in B can be partitioned into r subsequences V 1 , . . ., V r in a balanced manner with respect to the partial sums: For all n ∈ N, r, we have i k,v i ∈V v i -1 r i k v i 2.0005d. A similar bound holds for partitioning sequences of vector sets. Both results extend an earlier one of Bárány and Grinberg [I. Bárány, V.S. Grinberg, On some combinatorial questions in finite-dimensional spaces, Linear Algebra Appl. 41 (1981) 1-9] to partitions in arbitrarily many classes.

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## Abstract Let __V__~n~(q) denote a vector space of dimension __n__ over the field with __q__ elements. A set ${\cal P}$ of subspaces of __V__~n~(q) is a __partition__ of __V__~n~(q) if every nonzero element of __V__~n~(q) is contained in exactly one element of ${\cal P}$. Suppose there exists a p