  We study the presence of $L$-orthogonal elements in connection with Daugavet
centers and narrow operators. We prove that, if $\dens(Y)\leq \omega_1$ and
$G:X\longrightarrow Y$ is a Daugavet center, then $G(W)$ contains some
$L$-orthogonal for every non-empty $w^*$-open subset of $B_{X^{**}}$. In the
context of narrow operators, we show that if $X$ is separable and
$T:X\longrightarrow Y$ is a narrow operator, then given $y\in B_X$ and any
non-empty $w^*$-open subset $W$ of $B_{X^{**}}$ then $W$ contains some
$L$-orthogonal $u$ so that $T^{**}(u)=T(y)$. In the particular case that
$T^*(Y^*)$ is separable, we extend the previous result to $\dens(X)=\omega_1$.
Finally, we prove that none of the previous results holds in larger density
characters (in particular, a counterexample is shown for $\omega_2$ under
continuum hypothesis).
