1. Hedgehog theory via Euler calculus
- Author
-
Yves Martinez-Maure, Martinez-Maure, Yves, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Géométrie et Dynamique, and Institut Mathématique de Jussieu
- Subjects
index ,Pure mathematics ,Algebra and Number Theory ,Mixed volume ,Mathematical analysis ,2010 MSC: 28E99, 52A20, 52A30, 52A39, 53C65 ,Brunn-Minkowski theory ,Support function ,Euler calculus ,Minkowski addition ,mixed volumes ,Euler method ,symbols.namesake ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,Hedgehogs ,Euler characteristic ,Minkowski space ,symbols ,Euler integration ,Geometry and Topology ,convex bodies ,[MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG] ,Vector space ,Mathematics - Abstract
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.
- Published
- 2013