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.

On Reverses of the Blaschke-Santalo Inequality for Convex Bodies

Reisner proved a reverse of the Blaschke-Santal5 inequality for zonoid bodies, Bourgain and Milman showed another reverse of the Blaschke-Santal5 inequality for centered convex bodies. In this paper, two reverses of the Blaschke-Santal5 inequality for convex bodies are given by the Petty projection inequality and above two reverses. Further, using above methods, we also obtain two analogues of the Petty＇s conjecture for projection bodies, respectively.

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.

A Simple Approach to Estimating the Banach-Mazur Distance between Simplex and Convex Bodies

We use a simple approach to estimating the Banach-Mazur distance between convex bodies and simplex. As an application of this approach, we provide a purely analytic proof for the known result supСЕχn dBM（C,△）≤ n ＋ 2, where dBM（∵） denotes the Banach-Mazur distance, △ denotes an n-dimensional simplex and κ^n denotes the class of n-dimensional convex sets in R^n.

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.

Inclusion measures of convex bodies(Ⅱ)

In this paper, several inequalities for inclusion measures of convex bodies were obtained. The inclusion measure was proved to have concavity by considering the property of relative inner parallel body.

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.

Extreme properties of quermassintegrals of convex bodies

In this paper,we establish two theorems for the quermassintegrals of convex bodies,which are the generalizations of the well-known Aleksandrov's projection theorem and Loomis-Whitney's inequality,respectively.Applying these two theorems,we obtain a number of inequalities for the volumes of projections of convex bodies.Besides,we introduce the concept of the perturbation element of a convex body,and prove an extreme property of it.

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.

Some Inequalities for the L_p-polar Curvature Images of Star Bodies

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.

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.

Symmetry Function Induced by a Convex Body

This paper shows some properties of symmetry function induced by a convex body in a normal linear space. Some relationships between symmetry function induced by a convex body and Minkowski functional of the convex body are presented.