Rolf Schneider
It is proved that every convex body in the plane has a point such that the union of the body and its image under reflection in the point is convex. If the body is not centrally symmetric, then it has, in fact, three affinely independent points with this property.
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Borsuk number for planar convex bodies
本文利用图论工具刻画了Borsuk数等于2的平面凸体，给出了Borsuk数为3的实例，并证明欧氏球是唯一Borsuk数为3的中心对称平面凸体。
In this paper we study the bisections of a centrally symmetric planar convex body which minimize the maximum relative diameter functional. We give necessary and sufficient conditions for being a minimizing bisection, as well as analyzing the behavior of the so-called standard bisection.
Upper estimates of the diameter and the radius of the family of all planar convex bodies with respect to the Banach-Mazur distance are obtained. Namely, it is shown that the diameter does not exceed $\tfrac{19-\sqrt{73}}4\approx 2.614$, which improves the previously known bound of $3$, and that the radius does not exceed $\frac{117}{70}\approx 1.671$.
The purpose of this paper is to study the reflections of a convex body. In particular, we are interested in orthogonal reflections of its sections that can be extended to reflections of the whole body. For this reason, we need to study the case of a non-spherical ellipsoid, where a surprising structure arises (Section 2). These results allow us to give a new characterization of ellipsoids in terms of their reflections and, on the other hand, to prove a result deeply related to a conjecture due to K. Bezdek.
Cheeger sets for rotationally symmetric planar convex bodies
本文针对k-旋转对称平面凸体的Cheeger集展开研究，证明其具有k-旋转对称性且与原凸体的所有边都接触，丰富了凸几何中Cheeger集的性质研究。
On the minimal area of quadrangles circumscribed about planar convex bodies
研究证明平面凸体可包含在面积小于原凸体面积的四边形中，改进了Kuperberg的最佳上界。
Minkowski chirality: a measure of reflectional asymmetry of convex bodies
该研究提出闵可夫斯基手征性度量凸体反射不对称性，证明其与巴拿赫-马祖尔距离的不等式及连续性，确定平面三角形和平行四边形手征性的反射轴并得出$\sqrt{2}$为紧上界。
In this work we study the maximum relative diameter functional $d_M$ in the class of multi-rotationally symmetric planar convex bodies. A given set $C$ of this class is $k$-rotationally symmetric for $k\in\{k_1,\ldots,k_n\}\subset\nn$, and so it is natural to consider the standard $k_i$-partition $P_{k_i}$ associated to $C$ (which is a minimizing $k_i$-partition for $d_M$ when $k_i\geq3$) and the corresponding value $d_M(P_{k_i})$. We establish the relation among these values, characterizing the particular sets for which all these values coincide.
Planar radial mean bodies are convex
本文针对平面凸体的径向平均体凸性问题，证明了当参数p属于(-1,0)区间时，二维凸体的对应径向平均体仍保持凸性，补充了p≥0时凸性已知的相关结论。
An estimate of the centroid Banach-Mazur distance between planar convex bodies
本文研究了平面凸体在质心重合条件下的Banach-Mazur距离变体，证明了该距离上界为69/17。
In this work we study subdivisions of $k$-rotationally symmetric planar convex bodies that minimize the maximum relative diameter functional. For some particular subdivisions called $k$-partitions, consisting of $k$ curves meeting in an interior vertex, we prove that the so-called \emph{standard $k$-partition} (given by $k$ equiangular inradius segments) is minimizing for any $k\in\mathbb{N}$, $k\geq 3$. For general subdivisions, we show that the previous result only holds for $k\leq 6$. We also study the optimal set for this problem, obtaining that for each $k\in\mathbb{N}$, $k\geq 3$, it con…
Convex bodies with all characteristics planar
该研究针对辛空间ℝ²ⁿ（n>1）中的光滑强凸体，证明了满足所有特征线或所有外弹子球轨迹为平面这一条件的凸体，均为球的仿射辛映像。
Convex bodies with pairs of sections associated by reflections
本文证明了在特定条件下，两个凸体可以通过反射相互映射，其核心贡献是建立了超截面反射与整体反射之间的关系。
The Fourier transform of planar convex bodies and discrepancy over intervals of rotations
本文研究了平面凸体特征函数在仿射变换下平均的Fourier变换，建立了基于几何性质的上下界，并给出了仿射二次偏差渐近行为的精确结果，回答了Bilyk和Mastrianni的公开问题。
A strengthening of the Blaschke-Santaló inequality for $o$-symmetric planar convex bodies
本文加强了$o$-对称平面凸体的Blaschke-Santaló不等式，给出了具有最优阶次误差项的新不等式。
Area bounds for planar convex bodies containing a fixed number of interior integral points
本文研究包含固定数量内部整点的平面凸体的面积界问题，利用格宽数据证明面积界，并给出有理多边形依赖于分母的精确面积界及基于诺特公式的格多边形面积方程。
On Bezdek's conjecture for high-dimensional convex bodies with an aligned center of symmetry
本文在高维空间中验证了Bezdek猜想，假设通过固定点的截面具有反射轴，且互补不变子空间平行于固定超平面，证明了在正交和仿射设置中成立。
We extend the classical LR characterization of chirotopes of finite planar families of points to chirotopes of finite planar families of pairwise disjoint convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a chirotope of finite planar families of pairwise disjoint convex bodies if and only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the set of 3-subsets of J is a chirotope of finite planar families of pairwise disjoint convex bodies. Our main tool is the polarity map, i.e., the map that assigns to a convex body the set of lines missing its interi…
A centerpoint theorem for three planar convex bodies
本文证明在满足特定包含关系的三个平面凸体中，存在一点使任意闭半空间至少包含其总面积的2/9，从而以最优常数解决Oertel猜想的三切片情形。
The {\em minimal unfolded region} (or the {\em heart}) of a bounded subset $\Om$ in the Euclidean space is a subset of the convex hull of $\Om$ the definition of which is based on reflections in hyperplanes. It was introduced to restrict the location of the points that give extreme values of certain functions, such as potentials whose kernels are monotone functions of the distance, and solutions of differential equations to which Aleksandrov's reflection principle can be applied. %the temperature of a heat conductor with some initial-boundary condition, in which case the points are called the …
Homothetic packings of centrally symmetric convex bodies
本文研究中心对称凸体的位似堆积问题，证明随机半径位似堆积的接触图具有(2,2)-稀疏性，并证明存在余贫集使得任意(2,2)-稀疏平面图可实现为无应力位似堆积的接触图。
Exact Flatness Constant for One-Point Convex Bodies and the Discrete Isominwidth Problem: The Planar Case
研究整数规划中凸体最多含k个内格点时的最大格宽，证明平面凸体含至多一个内点时格宽不超过3，并导出等最小宽度不等式。
Given any polar pair of convex bodies we study its conjugate face maps and we characterize conjugate faces of non-exposed faces in terms of normal cones. The analysis is carried out using the positive hull operator which defines lattice isomorphisms linking three Galois connections. One of them assigns conjugate faces between the convex bodies. The second and third Galois connection is defined between the touching cones and the faces of each convex body separately. While the former is well-known, we introduce the latter in this article for any convex set in any finite dimension. We demonstrate…
We study upper bounds on the number of lattice points for convex bodies having their centroid at the origin. For the family of simplices as well as in the planar case we obtain best possible results. For arbitrary convex bodies we provide an upper bound, which extends the centrally symmetric case and which, in particular, shows that the centroid assumption is indeed much more restrictive than an assumption on the number of interior lattice points even for the class of lattice polytopes.
Bounding the diameter-width ratio using containment inequalities of means of convex bodies
本文研究了平面伪完全集的直径-宽度比，通过Minkowski中心凸集的包含不等式，定义了τ(K)并给出了其与闵可夫斯基不对称性相关的取值区域。
The semigroup of convex bodies in ${\mathbb R}^n$ with Minkowski addition has a canonical embedding into an abelian group; its elements have been called virtual convex bodies. Geometric interpretations of such virtual convex bodies have been particularly fruitful under the restriction to polytopes. For general convex bodies, mainly the planar case has been studied, as a part of the more general investigation of hedgehogs. Here we restrict ourselves to strictly convex bodies in ${\mathbb R}^n$. A particularly natural geometric interpretation of virtual convex bodies can then be seen in the set …
Area Minimization Among Group-Invariant Planar Convex Bodies of Constant Width
该研究在给定对称约束下，分类了固定原点的有限等距变换群G作用下不变的常宽度凸体中的最小面积者，明确了不同G对应的极小值类型。
Let $K \subset {\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\cdot |K^*| \ge 8$, with equality if and only if $K$ is a parallelogram. ($| \cdot |$ denotes volume). If $K \subset {\mathbb R}^2$ is a convex body, with $o \in {\text{int}}\,K$, then $|K|\cdot |K^*| \ge 27/4$, with equality if and only if $K$ is a triangle and $o$ is its centroid. If $K \subset {\mathbb R}^2$ is a convex body, then we have $|K| \cdot |[(K-K)/2)]^* | \ge 6$, with equality if and only if $K$ is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and…
On convex bodies with rotationally symmetric planar projections
本文证明：若凸体在所有二维平面上的正交投影均具有$q\ge4$重旋转对称性（至平移），则该凸体必为欧几里得球；$q=3$时存在非球形反例。
We show that for any Minkowski centered planar convex compact set $C$ the Harmonic mean of $C$ and $-C$ can be optimally contained in the arithmetic mean of the same sets if and only if the Minkowski asymmetry of $C$ is at most the golden ratio $(1+\sqrt{5})/2 \approx 1.618$. Moreover, the most asymmetric such set that is (up to a linear transformation) a special pentagon, which we call the golden house.
The covariogram g_K(x) of a convex body K \subseteq E^d is the function which associates to each x \in E^d the volume of the intersection of K with K+x. Matheron asked whether g_K determines K, up to translations and reflections in a point. Positive answers to Matheron's question have been obtained for large classes of planar convex bodies, while for d\geq 3 there are both positive and negative results. One of the purposes of this paper is to sharpen some of the known results on Matheron's conjecture indicating how much of the covariogram information is needed to get the uniqueness of determin…
On the total perimeter of disjoint convex bodies
本文通过引入基于法线距离的伪度量，简洁证明了Pinchasi定理：内部互不相交的$k$个凸体包含于周长为$p$、直径为$d$的凸体内时，其总周长不超过$p+2(k-1)d$。
Logarithmic Brunn--Minkowski Inequality under $n-2$ Reflection Symmetries
该研究证明了具有n-2个正交超平面对称性的原点对称凸体的对数Brunn-Minkowski不等式，讨论其等号情况与对数Minkowski问题的唯一性，并将不等式推广到特定对数凹测度。
This work is related to billiards and their applications in geometric optics. It is known that perfectly invisible bodies with mirror surface do not exist. It is natural to search for bodies that are, in a sense, close to invisible. We introduce a {\it visibility index} of a body measuring the mean angle of deviation of incident light rays, and derive a lower estimate to this index. This estimate is a function of the body's volume and of the minimal radius of a ball containing the body. This result is far from being final and opens a possibility for further research.
We prove that if two convex bodies $ K, L \subset \mathbb{R}^3$ satisfy the property that the orthogonal projections of $K$ and $L$ onto every plane containing the origin are roations of each other, then either $K$ and $L$ coincide or $L$ is the image of $K$ under a reflection about the origin
On generalized disc-polygons in plane convex bodies with a higher degree of smoothness
本文证明了平滑平面凸体中随机 $L$-凸多边形的顶点数期望与遗漏面积的幂级数展开式，扩展了已有渐近公式并推及 $L$-凸浮体与相对仿射表面积。
A characterization of the ellipsoid through planar grazes
本文通过平面掠射特征证明，若凸体满足特定条件，则其为椭球
A finite subset $K$ of $\mathbb{Z}^d$ is said to be lattice-convex if $K$ is the intersection of $\mathbb{Z}^d$ with a convex set. The covariogram $g_K$ of $K\subseteq \mathbb{Z}^d$ is the function associating to each $u \in \integer^d$ the cardinality of $K\cap (K+u)$. Daurat, Gérard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem on the reconstruction of lattice-convex sets $K$ from $g_K$. We provide a partial positive answer to this problem by showing that for $d=2$ and under mild extra assumptions, $g_K$ determines $K$ up to translations and reflections. As a com…
A generalization of pairwise intersecting Minkowski arrangement of centrally symmetric convex bodies is the pairwise intersecting Minkowski arrangement of order $μ$. Here, the homothetic copies of a centrally symmetric convex body are so that none of their interiors intersect the $μ$-kernel of any other. We give general upper and lower bounds on the cardinality of such arrangements, and study two special cases: For $d$-dimensional translates in classical pairwise intersecting Minkowski arrangement we prove that the sharp upper bound is $3^d$. For $μ=1$ the general version yields to another kno…
Minimal covering bodies and Brunn-Minkowski type inequalities for the covering radius
研究整数平移覆盖空间的极小凸体，证明其必为至少2d个面的多面体，并建立平面凸体Minkowski和覆盖半径的两个Brunn-Minkowski型不等式。
A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of $n$-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities, with one exception, for the four types of volumes regularly used in the literature: Busemann, Holmes-Thompson, and Gromov's mass and mass*. We examine the problem also for higher dimensi…
A pizza is a pair of planar convex bodies $A\subseteq B$,where $B$ represents the dough and $A$ the topping of the pizza. A partition of a pizza by straight lines is a succession of double operations:a cut by a full straight line, followed by a Euclidean move of one of theresulting pieces; then the procedure is repeated.The final partition is said to be fair if each resulting slice has the same amount of $A$ and the same amount of $B$.This note proves that, given an integer $n\geq2$, there exists a fair partition by straight lines of any pizza $(A,B)$ into $n$ parts if and onlyif $n$ is even.T…
We present a complete 3-dimensional Blaschke-Santaló diagram for planar convex bodies with respect to the four classical magnitudes inner and outer radius, diameter and (minimal) width in euclidean spaces.
In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction. We answer this question in the negative by constructing two infinitely smooth convex bodies of revolution about the $x_n$-axis in $\R^n$, $n\ge 3$, one origin symmetric and the other not centrally symmetric, with the same inner section function. Moreover, the pair of bodies can be arbitrarily close to the unit ball.
In the paper we investigate the continuity properties of the mapping $Φ$ which sends any non-empty compact connected hv-convex planar set $K$ to the associated generalized conic function $f_K$. The function $f_K$ measures the average taxicab distance of the points in the plane from the focal set $K$ by integration. The main area of the applications is the geometric tomography because $f_K$ involves the coordinate X-rays' information as second order partial derivatives \cite{NV3}. We prove that the Hausdorff-convergence implies the convergence of the conic functions with respect to both the sup…
The Andreev-Thurston Circle Packing Theorem is generalized to packings of convex bodies in planar simply connected domains. This turns out to be a useful tool for constructing conformal and quasiconformal mappings with interesting geometric properties. We attempt to illustrate this with a few results about uniformizations of finitely connected planar domains. For example, the following variation of a theorem by Courant, Manel and Shiffman is proved and generalized. If $G$ is an $n+1$-connected bounded planar domain, $H$ is a simply connected bounded planar domain, and $P_1,P_2,...,P_n$ are (co…
We propose strongly consistent algorithms for reconstructing the characteristic function 1_K of an unknown convex body K in R^n from possibly noisy measurements of the modulus of its Fourier transform \hat{1_K}. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K with its translates. In the many known situations in which the covariogram determines a convex…
We consider moments of the normalized volume of a symmetric or nonsymmetric random polytope in a fixed symmetric convex body. We investigate for which bodies these moments are extremized, and calculate exact values in some of the extreme cases. We show that these moments are maximized among planar convex bodies by parallelograms.
Mean distance between points inside and on the boundary of a convex body
本文研究扎波罗热茨-塔拉索夫猜想，针对中心对称凸体、平面凸体、外切凸体等不同情况，通过构造反例、推导公式等方式取得相关结论，还证明该猜想对三角形的逐投影形式成立。
The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that for d=2 the covariogram g_K determines K within the class of all planar convex bodies, up to translations and reflections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffrac…
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