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Alzheimer's disease is the most frequent type of dementia for elderly patients. Due to aging populations the occurrence of this disease will increase in the next years. Early diagnosis is crucial to be able to develop more powerful treatments. Brain perfusion changes can be a marker for Alzheimer's disease. In this article we study the use of SPECT perfusion imaging for the diagnosis of Alzheimer's disease differentiating between images from healthy subjects and images from Alzheimer's disease patients. Our classification approach is based on a linear programming formulation similar to the 1-norm support vector machines. In contrast with other linear hyperplane-based methods that perform simultaneous feature selection and classification, our proposed formulation incorporates proximity information about the features and generates a classifier that does not just select the most relevant voxels but the most relevant "areas" for classification resulting in more robust classifiers that are better suitable for interpretation. This approach is compared with the classical Fisher linear discriminant (FLD) classifier as well as with statistical parametric mapping (SPM). We tested our method on data from four European institutions. Our method achieved sensitivity of 84.4% at 90.9% specificity, this is considerable better the human experts. Our method also outperformed the ELD and SPM techniques. We conclude that our approach has the potential to be a useful help for clinicians.