Octahedrality and Gâteaux smoothness

Abstract

[EN] We prove that every Banach space admitting a Gâteaux smooth norm and containing a complemented copy of I1 has an equivalent renorming which is simultaneously Gâteaux smooth and octahedral. This is a partial solution to an open problem from the early nineties. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http://creativecommons .org /licenses /by -nc /4 .0/).The present work was done while the first author was on a PhD stay at the Department of Mathematics of the Faculty of Electrical Engineering at the Czech Technical University in Prague, and he is grateful for all the support and kindness received there. The authors thank the anonymous referee for his/her helpful comments that improved the quality of the manuscript. The first author was supported by Generalitat Valenciana (through Project PROMETEU/2021/070 and the predoctoral fellowship CIACIF/2021/378) and by MICIU/AEI/10.13039/501100011033 and by ERDF/EU (through Projects PID2019-105011GB-I00, PID2021-122126NB-C33 and PID2022-139449NB-I00). The second author was supported by GA23-04776S and SGS24/052/OHK3/1T/13 of CTU in Prague.Cobollo-Gómez, C.; Hájek, P. (2025). Octahedrality and Gâteaux smoothness. Journal of Mathematical Analysis and Applications. 543(2). https://doi.org/10.1016/j.jmaa.2024.128968543