Open sets D D in R N ( N ≥ 3 ) {R^N}\;(N \geq 3) with the property that D ¯ \bar D is a closed annulus { x : r 1 ≤ ‖ x ‖ ≤ r 2 } \{ x:{r_1} \leq \;\left \| x\right \| \; \leq {r_2}\} are characterized by quadrature formulae involving mean values of certain harmonic functions. One such characterization is used to give a criterion for the existence of a best harmonic L 1 {L^1} approximant to a function which is subharmonic (and satisfies some other conditions) in an annulus.