{"title":"Colorful positive bases decomposition and Helly-type results for cones","authors":"Grigory Ivanov","doi":"arxiv-2407.20961","DOIUrl":null,"url":null,"abstract":"We prove the following colorful Helly-type result: Fix $k \\in [d-1]$. Assume\n$\\mathcal{A}_1, \\dots, \\mathcal{A}_{d+(d-k)+1}$ are finite sets (colors) of\nnonzero vectors in $\\R^d$. If for every rainbow sub-selection $R$ from these\nsets of size at most $\\max \\{d+1, 2(d-k+1)\\}$, the system $\\langle {a},{x}\n\\rangle \\leq 0,\\; a \\in R$ has at least $k$ linearly independent solutions,\nthen at least one of the systems $\\langle {a},{x} \\rangle \\leq 0,\\; a \\in\n\\mathcal{A}_i,$ $i \\in [d+(d-k)+1]$ has at least $k$ linearly independent\nsolutions. A \\emph{rainbow sub-selection} from several sets refers to choosing at most\none element from each set (color). The Helly-number $\\max \\{d+1, 2(d-k+1)\\}$ and the number of colors\n$d+(d-k)+1$ are optimal. Our key observation is a certain colorful Carath\\'eodory-type result for\npositive bases.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.20961","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove the following colorful Helly-type result: Fix $k \in [d-1]$. Assume
$\mathcal{A}_1, \dots, \mathcal{A}_{d+(d-k)+1}$ are finite sets (colors) of
nonzero vectors in $\R^d$. If for every rainbow sub-selection $R$ from these
sets of size at most $\max \{d+1, 2(d-k+1)\}$, the system $\langle {a},{x}
\rangle \leq 0,\; a \in R$ has at least $k$ linearly independent solutions,
then at least one of the systems $\langle {a},{x} \rangle \leq 0,\; a \in
\mathcal{A}_i,$ $i \in [d+(d-k)+1]$ has at least $k$ linearly independent
solutions. A \emph{rainbow sub-selection} from several sets refers to choosing at most
one element from each set (color). The Helly-number $\max \{d+1, 2(d-k+1)\}$ and the number of colors
$d+(d-k)+1$ are optimal. Our key observation is a certain colorful Carath\'eodory-type result for
positive bases.