Geometric Study of Minkowski Differences of Plane Convex Bodies

Abstract

AbstractIn the Euclidean plane ℝ2, we define the Minkowski difference 𝒦–𝓛 of two arbitrary convex bodies 𝒦, 𝓛 as a rectifiable closed curve ℋh⊂ ℝ2that is determined by the differenceh=h𝒦–h𝓛of their support functions. This curve ℋhis called the hedgehog with support functionh. 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 ℝn+1, defined as (possibly singular and self-intersecting) hypersurfaces of ℝn+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 lengthmeasures and solve the extension of the Christoffel–Minkowski problemto 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 ℝ2and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.