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Geometric Study of Minkowski Differences of Plane Convex Bodies

  Published:2006-06-01
 Printed: Jun 2006
  • Yves Martinez-Maure
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Abstract

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_{\mathcal{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.
MSC Classifications: 52A30, 52A10, 53A04, 52A38, 52A39, 52A40 show english descriptions Variants of convex sets (star-shaped, ($m, n$)-convex, etc.)
Convex sets in $2$ dimensions (including convex curves) [See also 53A04]
Curves in Euclidean space
Length, area, volume [See also 26B15, 28A75, 49Q20]
Mixed volumes and related topics
Inequalities and extremum problems
52A30 - Variants of convex sets (star-shaped, ($m, n$)-convex, etc.)
52A10 - Convex sets in $2$ dimensions (including convex curves) [See also 53A04]
53A04 - Curves in Euclidean space
52A38 - Length, area, volume [See also 26B15, 28A75, 49Q20]
52A39 - Mixed volumes and related topics
52A40 - Inequalities and extremum problems
 

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