Abstract. In the Euclidean plane R 2 , we define the Minkowski difference K−Lof two arbitrary convex bodies K, L as a rectifiable closed curve H h ⊂ R 2 that is determined by the difference h = h K − h L of their support functions. This curve H h is called the hedgehog with support function h. More generally, the object of hedgehog theory is to study the Brunn-Minkowski theory in the vector space of Minkowski differences of arbitrary convex bodies of Euclidean space R n+1 , defined as (possibly singular and self-intersecting) hypersurfaces of R n+1 . Hedgehog theory is useful for: (i) studying convex bodies by splitting them into a sum in order to reveal their structure; (ii) converting analytical problems into geometrical ones by considering certain real functions as support functions. The purpose of this paper is to give a detailed study of plane hedgehogs, which constitute the basis of the theory. In particular: (i) we study their length measures and solve the extension of the Christoffel-Minkowski problem to plane hedgehogs; (ii) we characterize support functions of plane convex bodies among support functions of plane hedgehogs and support functions of plane hedgehogs among continuous functions; (iii) we study the mixed area of hedgehogs in R 2 and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.
Re  sume Â. Nous pre  sentons une extension partielle de line  galite  dAlexandrovFenchel aux he  rissons (enveloppes parame  tre  es par leur application de Gauss), et nous en de  duisons une se  rie dine  galite  s ge Âome  triques pour les he  rissons, dont une ine  galite  de type Brunn-Minkowski. Enfin, nous pre  sentons une application a Á le  tude du diagramme de Blaschke.Abstract. We present a partial extension of the Alexandrov-Fenchel inequality to hedgehogs (envelopes parametrized by their Gauss map), and we deduce a series of geometric inequalities for hedgehogs, one of which is an inequality of Brunn-Minkowski type. Finally, we present an application to the study of the Blaschke diagram.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.