We consider the Bayesian optimal design problem in the usual linear regression model. A version of Elfving’s Theorem is proved for a model robust Bayesian c-optimality criterion. The optimal design minimizes a weighted product where the factors are proportional to the expected posterior risks of the Bayesian estimators for the linear combinations of the parameters in different models. The geometric characterizations are used to state sufficient conditions which guarantee that the classical and the Bayesian optimal designs are supported at the same set of points or are identical. The Bayesian D-optimal design problem appears as a special case in this setup considering “nested” models and special linear combinations for the paramater of the “highest coefficients” in different models. Thus sufficient conditions on the precision matrices of the prior distribution are found that the Bayesian D-optimal and the classical optimal design are supported at the same set of points or are identical. The results are illustrated in several examples.