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Given a closed submanifold, or a compact regular domain, in Euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuation), and show that they give essentially the same result. We prove that some of these energies are invariant under MA¶bius transformations, thus giving a generalization to higher dimensions of the MA¶bius energy of knots.

We introduce different bases for the vector space of Sp(2)Sp(1)-invariant, translation invariant continuous valuations on the quaternionic plane and determine a complete set of kinematic formulas.