RECONSTRUCTING TRIANGLES INSCRIBED IN CONVEX BODIES FROM X-RAY FUNCTIONS

This paper provides a method using fixed-point theory for the reconstruction of the triangle inscribed in convex bodies from X-ray functions in three arbitrary mutual directions.

Inclusion measures of convex bodies （ Ⅰ ）

In this paper, the relations between inclusion measures of different bodies related to convex body K and the inclusion measure of convex body K itself were obtained.

Approximation of Convex Bodies by ConvexBodies

For the affine distance d(C,D) between two convex bodies C, D(?) Rn, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C, D) < n1/2 if one is an ellipsoid and another is symmetric, d(C, D) < n if both are symmetric, and from F. John's result and d(C1,C2) < d(C1,C3)d(C2,C3) one has d(C,D) < n2 for general convex bodies; M. Lassak proved d(C, D) < (2n - 1) if one of them is symmetric. In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asymmetry for convex bodies.

Asymptotic properties of two affine invariants for 1-unconditional convex bodies

Let K be a 1-unconditional convex bodies in Euclidean spaces.We study the asymptotic properties of two affine invariants m2（K） and S2（K） for a random simplex inside K.As an application,we discuss the asymptotic properties of two affine invariants m2（Bpn ） and S2（Bpn ）,where Bpn = {x ∈ Rn ： ‖x‖ p 1}.

On Properties of p-critical Points of Convex Bodies

Properties of the p-measures of asymmetry and the corresponding affine equivariant p-critical points, defined recently by the second author, for convex bodies are discussed in this article. In particular, the continuity of p-critical points with respect to p on （1, ＋∞） is confirmed, and the connections between general p-critical points and the Minkowski-critical points （∞-critical points） are investigated. The behavior of p-critical points of convex bodies approximating a convex bodies is studied as well.

Firey linear combinations of convex bodies

For convex bodies, the Firey linear combinations were introduced and studied in several papers. In this paper the mean width of the Firey linear combinations of convex bodies is studied, and the lower bound of the mean width of the Firey linear combinations of convex body and its polar body is given.

Stable Determination of Convex Bodies from Intersection Bodies

In this paper, by means of the dual Brunn-Minkowski theories and methods as well as the integral transform, we have established a stability version of nonsymmetric convex bodies from intersection bodies. In addition, a stability results of symmetric convex bodies from L_p-counterparts is established.

Determination of convex bodies from Γ-section functions

In this paper, we prove that any polygon P in R^2 containing a fixed smooth, strictly convex and origin-symmetric body Γ whose boundary is real analytic in its interior, can be determined by its Γ-section functions among the polygons.

A Characterization of the Polarity Mapping for Convex Bodies

In this paper,we establish a characterization of the polarity mapping for convex bodies for 1-dimensional convex bodies,which is a supplement to the result for such a characterization obtained by B?r?czky and Schneider.

Stability in the Shephard Problem for L_p-Projection of Convex Bodies

In this article, we study the convex bodies associated with Lp-projections in the Brunn-Minkowski-Firey theory, and apply the Fourier analytic methods to prove the linear stability in the Shephard problem for Lp-projections of convex bodies.

Quasi-Shephard's Problem on Projections of Convex Bodies

In this paper, we develop a Fourier analytic approach to study the problem in the Brunn-Minkowski-Firey theory of convex bodies. We formulate and solve a quasi-Shephard＇s problem on projections of convex bodies.

p-Minkowski Type Measures of Asymmetry for Convex Bodies

In this paper,the p-Minkowski type measures of asymmetry for convex bodies,which have the well-known Minkowski measure of asymmetry as the special case p=∞,are introduced,and some properties of the p-Minkowski type measures are investigated.

ON MEAN CURVATURES OF A PARALLEL CONVEX BODY

In this article, we obtain some results about the mean curvature integrals of the parallel body of a convex set in R^n. These mean curvature integrals are generalizations of the Santalo＇s results.

Radial mean bodies of simplices

In this paper, it is proved that the radial pth mean body RpK（p 〉 0） is homothetic to the difference body DK when K is a simplex, Furthermore, the equality Rp（Rq） = Rq（Rp） is established when p 〉 0 and q 〉 0. It is also proved the Brunn-Minkowski inequality of radial pth mean body of simplex and uniqueness property.

L_p-centroid Bodies and Its Characterizations

In this paper, we study the characteristic properties for Lp-centroid bod- ies, and an improved version of Busemann-Petty problem for Lp-centroid bodies is obtained. In addition, using the definitions of Lp-pole curvature image and Lp-affine surface area, a new proof of Busemann-Petty problem for Lp-centroid bodies is given.

Duality in interaction potentials for curved surface bodies and inside particles

Based on the viewpoint of duality, this paper studies the interaction between a curved surface body and an inside particle. By convex/concave bodies with geometric duality, interaction potentials of particles located outside and inside the curved surface bodies are shown to have duality. With duality, the curvature-based potential between a curved surface body and an inside particle is derived. Furthermore, the normal and tangential driving forces exerted on the particle are studied and expressed as a function of curvatures and curvature gradients. Numerical experiments are designed to test accuracy of the curvature-based potential.

NORMALLY DISTRIBUTED PROBABILITY MEASURE ON THE METRIC SPACE OF NORMS

In this paper we propose a method to construct probability measures on the space of convex bodies. For this purpose, first, we introduce the notion of thinness of a body. Then we show the existence of a measure with the property that its pushforward by the thinness function is a probability measure of truncated normal distribution. Finally, we improve this method to find a measure satisfying some important properties in geometric measure theory.

关于L_p-极曲率映像的几个不等式（英文）

Zhu,Lü and Leng extended the concept of L_p-polar curvature image. We continuously study the L_p-polar curvature image and mainly expound the relations between the volumes of star bodies and their L_p-polar curvature images in this article. We first establish the L_p-affine isoperimetric inequality associated with L_p-polar curvature image. Secondly,we give a monotonic property for L_p-polar curvature image. Finally, we obtain an interesting equation related to L_p-projection body of L_p-polar curvature image and L_p-centroid body.

SOME DUAL KINEMATIC FORMULAS

In this article, some kinematic formulas for dual quermassintegral of star bodies and for chord power integrals of convex bodies are established by using dual mixed volumes. These formulas are the extensions of the fundamental kinematic formula involving quermassintegral to the case of dual quermassintegral and chord power integrals.

LOOMIS-WHITNEY'S INEQUALITY ABOUT MIXED BODY

In this article, the authors establish two theorems for mixed body, which are the generalizations of the well-known Loomis-Whitney＇s inequality.