$L$-orthogonality, octahedrality and Daugavet property in Banach spaces

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 [9] is negative. Furthermore, we prove that the abundance of almost $L$-orthogonal vectors in a Banach space $X$ (almost Daugavet property) whose density character is $\omega_1$ implies the abundance of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$. In fact, we get that a Banach space $X$ whose density character is $\omega_1$ verifies the Daugavet property if, and only if, the set of vectors in $X^{**}$ being $L$-orthogonal to $X$ is weak-star dense in $X^{**}$. We also prove that, under CH, the previous characterisation is false for Banach spaces with larger density character. Finally, some consequences on Daugavet property in the setting of $L$-embedded spaces are obtained.