This is an announcement for the paper “$L$-orthogonality, octahedrality and Daugavet property in Banach spaces” by Ginés López-Pérezhttps://arxiv.org/search/math?searchtype=author&query=L%C3%B3pez-P%C3%A9rez%2C+G, Abraham Rueda Zocahttps://arxiv.org/search/math?searchtype=author&query=Zoca%2C+A+R.

Abstract: We prove that the abundance of almost $L$-orthogonal vectors in a Banach space $X$ (almost Daugavet property) implies the abundance of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$. In fact, we get that a Banach space $X$ verifies the Daugavet property if, and only if, the set of vectors in $X^{**}$ being $L$-orthogonal to $X$ is weak-star dense in $X^{**}$. In contrast with the separable case, we prove that the existence of almost $L$-orthogonal vectors in a nonseparable Banach space $X$ (octahedrality) does not imply the existence of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$, which shows that the answer to an environment question in [7] is negative. Also, in contrast with the separable case, we obtain that the existence of almost $L$-orthogonal vectors in a nonseparable Banach space $X$ (octahedrality) does not imply the abundance of almost $L$-orthogonal vectors in Banach space $X$ (almost Daugavet property), which solves an open question in [21]. Some consequences on Daugavet property in the setting of $L$-embedded spaces are also obtained.

https://arxiv.org/abs/1912.09039