Reconstruction of Convex Bodies from Projection Curvature Radius Function

Some extensions of the class of $k$-convex bodies

We study relations of some classes of $k$-convex, $k$-visible bodies in Euclidean spaces. We introduce and study \textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular, \textrm{$k$-circular convex} and \textrm{$k$-circular visible} ones. Investigation of these bodies more general than $k$-convex and $k$-visible ones allows us to generalize some classical results of geometric tomography and find their new applications.

Convolutions and multiplier transformations of convex bodies

Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Min…

General Lp affine isoperimetric inequalities

Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann--Petty centroid inequality.

Isoperimetric inequalities in convex cylinders and cylindrically bounded convex bodies

In this paper we consider the isoperimetric profile of convex cylinders $K\times\mathbb{R}^q$, where $K$ is an $m$-dimensional convex body, and of cylindrically bounded convex sets, i.e, those with a relatively compact orthogonal projection over some hyperplane of $\mathbb{R}^{n+1}$, asymptotic to a right convex cylinder of the form $K\times\mathbb{R}$, with $K\subset\mathbb{R}^n$. Results concerning the concavity of the isoperimetric profile, existence of isoperimetric regions, and geometric descriptions of isoperimetric regions for small and large volumes are obtained.

关于逆投影不等式

On the Reverse Projection Inequality

本文刻画了三维一般凸体逆投影不等式的等号情形，证明该不等式在n≥9的所有维度均不成立，反例为至多n+2个面的满维多面体。

A sausage body is a unique solution for a reverse isoperimetric problem

We consider the class of $λ$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/λ$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius $1/λ$ (a sausage body) is a unique volume minimizer among all $λ$-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve…

Bezdek猜想的一个解

A solution to Bezdek's conjecture

本文证明在n≥2时，给定内径的λ-凸体中λ-凸透镜平均宽度最大，肯定回答Bezdek猜想；附加对称性时解决其任意内蕴体积的类似猜想；还证明固定外接半径的λ-凸体中λ-凸纺锤体平均宽度唯一最小。

Reverse isoperimetric problems under curvature constraints

In this paper we solve several reverse isoperimetric problems in the class of $λ$-convex bodies, i.e., convex bodies whose curvature at each point of their boundary is bounded below by some $λ> 0$. We give an affirmative answer in $\mathbb{R}^3$ to a conjecture due to Borisenko which states that the $λ$-convex lens, i.e., the intersection of two balls of radius $1/λ$, is the unique minimizer of volume among all $λ$-convex bodies of given surface area. Also, we prove a reverse inradius inequality: in model spaces of constant curvature and arbitrary dimension, we show that the $λ$-convex lens (p…

平面中反等周不等式的稳定性：面积、Cheeger和内半径

Stability of reverse isoperimetric inequalities in the plane: area, Cheeger, and inradius

本文研究了二维平面中反等周不等式的稳定性，证明了$\lambda$-凸体的反等周不等式稳定性，并提出了新的反Cheeger不等式结果。

平坦环面上凸体的长度正交谱

Length orthospectrum of convex bodies on flat tori

类比负曲率流形Pollicott-Ruelle共振研究，定义适配平坦环面平移不变Finsler度量测地向量场的各向异性Sobolev空间，研究凸体正交测地线性质，定义几何Epstein函数并证其亚纯延拓、算留数，证正交测地线长度与磁Laplacian谱的Poisson型求和公式。