A matrix version of the Steinitz lemma
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914673142530048 |
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| author | Barany, Imre |
| author_facet | Barany, Imre |
| contents | The Steinitz lemma, a classic from 1913, states that $a_1,\ldots,a_n$, a sequence of vectors in $\R^d$ with $\sum_1^n a_i=0$, can be rearranged so that every partial sum of the rearranged sequence has norm at most $2d\max \|a_i\|$. In the matrix version $A$ is a $k\times n$ matrix with entries $a_i^j \in \R^d$ with $\sum_{j=1}^k\sum_{i=1}^na_i^j=0$. It is proved in \cite{OPW} that there is a rearrangement of row $j$ of $A$ (for every $j$) such that the sum of the entries in the first $m$ columns of the rearranged matrix has norm at most $40d^5\max \|a_i^j\|$ (for every $m$). We improve this bound to $(4d-2)\max \|a_i^j\|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10102 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A matrix version of the Steinitz lemma Barany, Imre Combinatorics primary 05A20, secondary 52A22 The Steinitz lemma, a classic from 1913, states that $a_1,\ldots,a_n$, a sequence of vectors in $\R^d$ with $\sum_1^n a_i=0$, can be rearranged so that every partial sum of the rearranged sequence has norm at most $2d\max \|a_i\|$. In the matrix version $A$ is a $k\times n$ matrix with entries $a_i^j \in \R^d$ with $\sum_{j=1}^k\sum_{i=1}^na_i^j=0$. It is proved in \cite{OPW} that there is a rearrangement of row $j$ of $A$ (for every $j$) such that the sum of the entries in the first $m$ columns of the rearranged matrix has norm at most $40d^5\max \|a_i^j\|$ (for every $m$). We improve this bound to $(4d-2)\max \|a_i^j\|$. |
| title | A matrix version of the Steinitz lemma |
| topic | Combinatorics primary 05A20, secondary 52A22 |
| url | https://arxiv.org/abs/2308.10102 |