The determination of an optimal design for a given regression problem is an intricate optimization problem, especially for models with multivariate predictors. Design admissibility and invariance are main tools to reduce the complexity of the optimization problem and have been successfully applied for models with univariate predictors. In particular several authors have developed sufficient conditions for the existence of saturated designs in univariate models, where the number of support points of the optimal design equals the number of parameters. These results generalize the celebrated de la Garza phenomenon (de la Garza, 1954) which states that for a polynomial regression model of degree $p-1$ any optimal design can be based on at most $p$ points. This paper provides - for the first time - extensions of these results for models with a multivariate predictor. In particular we study a geometric characterization of the support points of an optimal design to provide sufficient conditions for the occurrence of the de la Garza phenomenon in models with multivariate predictors and characterize properties of admissible designs in terms of admissibility of designs in conditional univariate regression models.