Solving the Gleason problem on linearly convex domains

Let V be a bounded, connected linearly convex set in C^n with $C^{1+\epsilon}$-boundary. We show that the maximal ideal (both in A(V) and $H^{\infty}(V)$) consisting of all functions vanishing at p in V is generated by the coordinate functions z_1 - p_1, ..., z_n - p_n.