Hedgehog theory via Euler calculus

Yves Martinez-Maure (1)

1 IMJ - Institut de Mathématiques de Jussieu

Hedgehogs are (possibly singular and self-intersecting) hypersurfaces that describe Minkowski di¤erences of convex bodies in 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. There is a close relationship between Minkowski addition and convolution with respect to the Euler characteristic. In this paper, we extend it 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. Finally, we present some applications.

Mots clés: Hedgehogs; convex bodies; Brunn-Minkowski theory; Euler characteristic; Euler integration; index; mixed volumes

Domaines: Géométrie différentielle [math.DG]

Yves Martinez-Maure. Hedgehog theory via Euler calculus. 2013. ⟨hal-00776724v2⟩