  It is shown by a counterexample that isocapacitary and isoperimetric
constants of a multi-dimensional Euclidean domain starshaped with respect to a
ball are not equivalent. Sharp integral inequalities involving the harmonic
capacity which imply Faber-Krahn property of the fundamental Dirichlet-Laplace
eigenvalue are obtained. Necessary and sufficient conditions ensuring integral
inequalities between a difference seminorm and the $L_p$-norm of the gradient
are found.
