A new geometric approach to data analysis using the Minkowski polytope

We introduce a new approach to the analysis of random samples in Rd, based on a geometric transformation called the "Minkowski polytope" (MP). We describe how a theorem by Minkowski guarantees the existence and uniqueness of such a transformation, discuss its construction, and state a result about the almost sure convergence of a scaled version of the MP in R2. We show how the shape of the MP is sensitive to the presence of outliers and correlation in the sample. Finally, we use the MP to develop a new Monte Carlo test for spatial randomness over non-uniform populations, and illustrate its application on a well-known dataset of leukemia cases in the state of New York.