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Supporting vectors for the \(\ell _1\)-norm and the \(\ell _{\infty }\)-norm and an application

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Abstract

A supporting vector of a matrix A for a certain norm \(\Vert \cdot \Vert \) on \(\mathbb {R}^n\) is a vector x such that \(\Vert x\Vert =1\) and \(\Vert Ax\Vert =\Vert A\Vert =\displaystyle \max _{\Vert y\Vert =1}\Vert Ay\Vert \). In this manuscript, we characterize the existence of supporting vectors in the infinite-dimensional case for both the \(\ell _1\)-norm and the \(\ell _{\infty }\)-norm. Besides this characterization, our theorems provide a description of the set of supporting vectors for operators on \(\ell _{\infty }\) and \(\ell _1\). As an application of our results in the finite-dimensional case for both the \(\ell _1\)-norm and the \(\ell _{\infty }\)-norm, we study meteorological data from stations located on the province of Cádiz (Spain). For it, we consider a matrix database with the highest temperature deviations of these stations.

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Data availibility statement

The data that support the findings of this study are openly available in Meteorological Agency of Spain (AEMET) at https://opendata.aemet.es/centrodedescargas/inicio.

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Author notes

  1. Alberto Sánchez-Alzola, Francisco Javier García-Pacheco, Enrique Naranjo-Guerra and Soledad Moreno-Pulido: All authors contributed equally to the work.

Authors and Affiliations

  1. Department of Statistics and Operation Research, College of Engineering, University of Cadiz, Puerto Real, 11510, Spain

    Alberto Sánchez-Alzola & Enrique Naranjo-Guerra

  2. Department of Mathematics, College of Engineering, University of Cadiz, Puerto Real, 11510, Spain

    Francisco Javier García-Pacheco & Soledad Moreno-Pulido

Authors
  1. Alberto Sánchez-Alzola
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  2. Francisco Javier García-Pacheco
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  3. Enrique Naranjo-Guerra
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  4. Soledad Moreno-Pulido
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Corresponding author

Correspondence to Francisco Javier García-Pacheco.

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This work has been supported by the Research Grant PGC-101514-B-I00 awarded by the Spanish Ministry of Science, Innovation and Universities and partially funded by FEDER. This work has also been co-financed by the 2014–2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia. Project reference: FEDER-UCA18-105867.

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Sánchez-Alzola, A., García-Pacheco, F.J., Naranjo-Guerra, E. et al. Supporting vectors for the \(\ell _1\)-norm and the \(\ell _{\infty }\)-norm and an application. Math Sci 15, 173–187 (2021). https://doi.org/10.1007/s40096-021-00400-w

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  • DOI: https://doi.org/10.1007/s40096-021-00400-w

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