Geometric Study of Minkowski Differences of Plane Convex Bodies

Abstract In the Euclidean plane ℝ 2 , we define the Minkowski difference 𝒦–𝓛 of 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 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 ℝ 2 and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.

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