Metric entropy for functions of bounded total generalized variation

We establish a sharp estimate for a minimal number of binary digits (bits) needed to represent all bounded total generalized variation functions taking values in a general totally bounded metric space (E, ρ ) up to an accuracy of ∊ > 0 with respect to the L1-distance. Such an estimate is explicitly computed in terms of doubling and packing dimensions of (E, ρ ). The obtained result is applied to provide an upper bound on the metric entropy for a set of entropy admissible weak solutions to scalar conservation laws in one-dimensional space with weakly genuinely nonlinear fluxes.s