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The determination of an optimal design for a given regression problem is anintricate optimization problem, especially for models with multivariatepredictors. Design admissibility and invariance are main tools to reduce thecomplexity of the optimization problem and have been successfully applied formodels with univariate predictors. In particular several authors have developedsufficient conditions for the existence of saturated designs in univariatemodels, where the number of support points of the optimal design equals thenumber of parameters. These results generalize the celebrated de la Garzaphenomenon (de la Garza, 1954) which states that for a polynomial regressionmodel 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 formodels with a multivariate predictor. In particular we study a geometriccharacterization of the support points of an optimal design to providesufficient conditions for the occurrence of the de la Garza phenomenon inmodels with multivariate predictors and characterize properties of admissibledesigns in terms of admissibility of designs in conditional univariateregression models.

Design admissibility and de la Garza phenomenon in multi-factor experiments