Regularity of absolute minimizers for continuous convex Hamiltonians

For any $n\ge 2$, $\Omega\subset\rn$, and any given convex and coercive Hamiltonian function $H\in C^{0}(\rn)$, we find an optimal sufficient condition on $H$, that is, for any $c\in\mathbb R$, the level set $H^{-1}(c)$ does not contains any line segment, such then any absolute minimizer $u\in AM_H(\Omega)$ enjoys the linear approximation property. As consequences, we show that when $n=2$, if $u\in AM_H(\Omega)$ then $u\in C^1$; and if $u\in AM_H(\rr^2)$ satisfies a linear growth at the infinity, then $u$ is a linear function on $\rr^2$. In particular, if $H$ is a strictly convex Banach norm $\|\cdot\|$ on $\mathbb R^2$, e.g. the $l_\alpha$-norm for $1<\alpha<1$, then any $u\in AM_H(\Omega)$ is $C^1$. The ideas of proof are, instead of PDE approaches, purely variational and geometric.