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1. CJM 2013 (vol 67 pp. 3)
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)
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 |