Size Theory was proposed in the early 90's as a geometrical/topological approach to the problem of Shape Comparison, a very lively research topic in the fields of Computer Vision and Pattern Recognition. The basic idea is to discriminate shapes by comparing shape properties that are provided by continuous functions valued in R, called measuring functions and defined on topological spaces associated to the objects to be studied. In this way, shapes can be compared by using a descriptor named

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... function, whose role is to capture the features described by measuring functions and represent them in a quantitative way. However, a common scenario in applications is to deal with multidimensional information. This observation has led to considering vector-valued measuring functions, and consequently the multidimensional extension of size functions, namely the k-dimensional size functions. In this work we survey some recent results about size functions in this multidimensional setting, with particular reference to the localization of their discontinuities.