The Steinitz lemma in \(\ell^ 2_{\infty{}}\). (English) Zbl 0745.52001
Let \(a,b\geq 1\) such that \(a+b\geq 3\) and let \(u_ 1,\dots,u_ n\) be vectors in the rectangle \(| x| \leq 1\), \(| y| \leq 1\) in \(\mathbb{R}^ 2\) for which \(u_ 1+\dots+u_ n=0\). Then there is a rearrangement \(v_ 1,\dots,v_ n\) of \(u_ 1,\dots,u_ n\) such that all partial sums \(v_ 1,v_ 1+v_ 2,\dots,v_ 1+\dots+v_ n\) are contained in the rectangle \(| x| \leq a\), \(| y|\leq b\). The first result of this kind was given by E. Steinitz [J. Reine Angew. Math. 143, 128-175 (1913)], more recent ones are due to Bárány, Grinberg and the author.
Reviewer: P.Gruber (Wien)
MSC:
| 52A10 | Convex sets in \(2\) dimensions (including convex curves) |
| 05A05 | Permutations, words, matrices |
References:
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