Geometric Study of Minkowski Differences of Plane Convex Bodies

Martinez-Maure, Yves

Canadian journal of mathematics, Tome 58 (2006) no. 3, pp. 600-624

Résumé

In the Euclidean plane ${{\mathbb{R}}^{2}}$ , we define the Minkowski difference $\mathcal{K}-\mathcal{L}$ of two arbitrary convex bodies $\mathcal{K},\mathcal{L}$ as a rectifiable closed curve ${{\mathcal{H}}_{h}}\subset {{\mathbb{R}}^{2}}$ that is determined by the difference $h={{h}_{K}}-{{h}_{\mathcal{L}}}$ of their support functions. This curve ${{\mathcal{H}}_{h}}$ 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 ${{\mathbb{R}}^{n+1}}$ , defined as (possibly singular and self-intersecting) hypersurfaces of ${{\mathbb{R}}^{n+1}}$ . 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 ${{\mathbb{R}}^{2}}$ and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.

Détail

DOI : 10.4153/CJM-2006-025-x

Mots-clés : 52A30, 52A10, 53A04, 52A38, 52A39, 52A40

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