Certainty relations, mutual entanglement and non-displacable manifolds

{We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size N in L orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For L=2 the maximal average entropy saturates at logN as there exists a mutually coherent state, but certainty relations are shown to be nontrivial for L≥3 measurements. In the case of a prime power dimension