Regularity of absolute minimizers for continuous convex Hamiltonians

For any n≥2, Ω⊂Rⁿ, and any given convex and coercive Hamiltonian function H∈C⁰(Rⁿ), we find an optimal sufficient condition on H, that is, for any c∈R, the level set H⁻¹(c) does not contain any line segment, such then any absolute minimizer u∈AM_H(Ω) enjoys the linear approximation property. As consequences, we show that when n=2, if u∈AM_H(Ω) then u∈C¹; and if u∈AM_H(R²) satisfies a linear growth at the infinity, then u is a linear function on R². In particular, if H is a strictly convex Banach norm ‖⋅‖ on R², e.g. the l_α-norm for 1<α<∞, then any u∈AM_H(Ω) is C¹. The ideas of proof are, instead of PDE approaches, purely variational and geometric.