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Search: MSC category 52A38 ( Length, area, volume [See also 26B15, 28A75, 49Q20] )

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1. CJM 2013 (vol 67 pp. 3)

Alfonseca, M. Angeles; Kim, Jaegil
On the Local Convexity of Intersection Bodies of Revolution
One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach-Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

Keywords:convex bodies, intersection bodies of star bodies, Busemann's theorem, local convexity
Categories:52A20, 52A38, 44A12

2. CJM 2012 (vol 66 pp. 700)

He, Jianxun; Xiao, Jinsen
Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two
Let $F_{2n,2}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf P$ denote the affine automorphism group of $F_{2n,2}$. In this article the theory of continuous wavelet transform on $F_{2n,2}$ associated with $\mathbf P$ is developed, and then a type of radial wavelets is constructed. Secondly, the Radon transform on $F_{2n,2}$ is studied and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform, the others are from group Fourier transform. By using wavelet transform we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. Specially, if $n=1$, $F_{2,2}$ is the $3$-dimensional Heisenberg group $H^1$, the inversion formula of the Radon transform is valid which is associated with the sub-Laplacian on $F_{2,2}$. This result cannot be extended to the case $n\geq 2$.

Keywords:Radon transform, wavelet transform, free nilpotent Lie group, unitary representation, inversion formula, sub-Laplacian
Categories:43A85, 44A12, 52A38

3. CJM 2012 (vol 65 pp. 1401)

Zhao, Wei; Shen, Yibing
A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results
In this paper, we establish a universal volume comparison theorem for Finsler manifolds and give the Berger-Kazdan inequality and Santaló's formula in Finsler geometry. Being based on these, we derive a Berger-Kazdan type comparison theorem and a Croke type isoperimetric inequality for Finsler manifolds.

Keywords:Finsler manifold, Berger-Kazdan inequality, Berger-Kazdan comparison theorem, Santaló's formula, Croke's isoperimetric inequality
Categories:53B40, 53C65, 52A38

4. CJM 2006 (vol 58 pp. 600)

Martinez-Maure, Yves
Geometric Study of Minkowski Differences of Plane Convex Bodies
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.

Categories:52A30, 52A10, 53A04, 52A38, 52A39, 52A40

5. CJM 1999 (vol 51 pp. 449)

Bahn, Hyoungsick; Ehrlich, Paul
A Brunn-Minkowski Type Theorem on the Minkowski Spacetime
In this article, we derive a Brunn-Minkowski type theorem for sets bearing some relation to the causal structure on the Minkowski spacetime $\mathbb{L}^{n+1}$. We also present an isoperimetric inequality in the Minkowski spacetime $\mathbb{L}^{n+1}$ as a consequence of this Brunn-Minkowski type theorem.

Keywords:Minkowski spacetime, Brunn-Minkowski inequality, isoperimetric inequality
Categories:53B30, 52A40, 52A38

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