Hedgehog theory via Euler calculus

Hedgehogs are (possibly singular and self-intersecting) hypersurfaces that describe Minkowski differences of convex bodies in $$\mathbb {R}^{n+1}$$ . They are the natural geometrical objects when one seeks to extend parts of the Brunn–Minkowski theory to a vector space which contains convex bodies. In terms of characteristic functions, Minkowski addition of convex bodies correspond to convolution with respect to the Euler characteristic. In this paper, we extend this relationship to hedgehogs with an analytic support function. In this context, resorting only to the support functions and the Euler characteristic, we give various expressions for the index of a point with respect to a hedgehog.