In the first part of this paper, the authors prove results that characterize a closed ball or annulus in \(\underset \tilde{} R^ N\) \((N\geq 3)\), in terms of the mean values of harmonic functions. As an example, we quote Theorem 2.3(i), in which \(\lambda\) denotes Lebesgue measure, M(h,r) denotes the usual mean value of a harmonic function h over a sphere of radius r centred at the origin, and a domain \(\omega\) is said to be locally connected to \(\infty\) if there is a piecewise linear, continuous function \(\phi: [0,\infty [\to \omega\) such that \(\| \phi (t)\| \to \infty\) as \(t\to \infty:\)
If D is a nonempty open set in \(\underset \tilde{} R^ N\) such that \(\lambda(\bar D)<\infty\) and D has no component that is locally connected to \(\infty\), and if \(\lambda(\bar D)^{-1}\int_{\bar D}h d\lambda =M(h,r)\) for every function h harmonic on \(\underset \tilde{} R^ N\), then \(\bar D\) is a closed ball of centre 0.
This improves a result of M. Goldstein, W. Haussmann and L. Rogge [Trans. Am. Math. Soc. 305, No.2, 505-515 (1988; Zbl 0667.31004)]. In the second part of the paper, Theorem 2.3 is used to prove the existence, under certain conditions, of a best harmonic \(L^ 1\) approximation to a subharmonic function on an annulus. The analogous result for the unit ball was given in the aforementioned paper by M. Goldstein et al.