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Geometric Study of Minkowski Differences of Plane Convex Bodies


Published:20060601
Printed: Jun 2006
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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 BrunnMinkowski theory in the vector space of
Minkowski differences of arbitrary convex bodies of Euclidean space $\mathbb{R}
^{n+1}$, defined as (possibly singular and selfintersecting) 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
ChristoffelMinkowski 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 (starshaped, ($m, n$)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 (starshaped, ($m, n$)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
