1. CJM 2016 (vol 69 pp. 873)
 Xiao, Jie; Ye, Deping

Anisotropic Sobolev Capacity with Fractional Order
In this paper, we introduce the anisotropic
Sobolev capacity with fractional order and develop some basic
properties for this new object. Applications to the theory of
anisotropic fractional Sobolev spaces are provided. In particular,
we give geometric characterizations for a nonnegative Radon
measure $\mu$ that naturally induces an embedding of the anisotropic
fractional Sobolev class $\dot{\Lambda}_{\alpha,K}^{1,1}$ into
the $\mu$basedLebesguespace $L^{n/\beta}_\mu$ with $0\lt \beta\le
n$. Also, we investigate the anisotropic fractional $\alpha$perimeter.
Such a geometric quantity can be used to approximate the anisotropic
Sobolev capacity with fractional order. Estimation on the constant
in the related Minkowski inequality, which is asymptotically
optimal as $\alpha\rightarrow 0^+$, will be provided.
Keywords:sharpness, isoperimetric inequality, Minkowski inequality, fractional Sobolev capacity, fractional perimeter Categories:52A38, 53A15, 53A30 

2. 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 BanachMazur
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 

3. 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 subLaplacian 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, subLaplacian Categories:43A85, 44A12, 52A38 

4. 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 BergerKazdan inequality and
SantalÃ³'s formula in Finsler geometry. Being based on these, we
derive a BergerKazdan type comparison theorem and a Croke type
isoperimetric inequality for Finsler manifolds.
Keywords:Finsler manifold, BergerKazdan inequality, BergerKazdan comparison theorem, SantalÃ³'s formula, Croke's isoperimetric inequality Categories:53B40, 53C65, 52A38 

5. CJM 2006 (vol 58 pp. 600)
 MartinezMaure, 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 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.
Categories:52A30, 52A10, 53A04, 52A38, 52A39, 52A40 

6. CJM 1999 (vol 51 pp. 449)
 Bahn, Hyoungsick; Ehrlich, Paul

A BrunnMinkowski Type Theorem on the Minkowski Spacetime
In this article, we derive a BrunnMinkowski 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
BrunnMinkowski type theorem.
Keywords:Minkowski spacetime, BrunnMinkowski inequality, isoperimetric inequality Categories:53B30, 52A40, 52A38 
