Summary: This paper describes the characterization of optimal constants for some coercivity inequalities in \(W^{1,p}(\Omega)\), \(1< p < \infty\). A general result involving inequalities of \(p\)-homogeneous forms on a reflexive Banach space is first proved. The constants are shown to be the least eigenvalues of certain eigenproblems with equality holding for the corresponding eigenfunctions. This result is applied to three different classes of coercivity results on \(W^{1,p}(\Omega)\). The inequalities include very general versions of the Friedrichs and Poincaré inequalities. Scaling laws for the inequalities are also described.