We propose a new method for selecting a common subset of explanatory variables where the aim is to model several response variables. The idea is a natural extension of the LASSO technique proposed by Tibshirani (1996) and is based on the (joint) residual sum of squares while constraining the parameter estimates to lie within a suitable polyhedral region. The properties of the resulting convex programming problem are analyzed for the special case of an orthonormal design. For the general case, we develop an efficient interior point algorithm. The method is illustrated on a dataset with infrared spectrometry measurements on 14 qualitatively different but correlated responses using 770 wavelengths. The aim is to select a subset of the wavelengths suitable for use as predictors for as many of the responses as possible.