We study the control of the familywise error rate (FWER) in the linear Gaussian model when the n × p design matrix is of rank p. A procedure based on a lasso-type estimator is optimized with respect to the volume of the acceptance region. Surprisingly, it reduces to a simple new multiple testing procedure based on the maximum likelihood estimator. This new procedure is shown to be more powerful than the existing ones. Numerical experiments highlight the performances of our approach compared to the state-of-the-art procedures. An application to the detection of metabolites in metabolomics is provided.