Richard J. Gardner
We present a method which shows that in $\Eb$ the Busemann-Petty problem, concerning central sections of centrally symmetric convex bodies, has a positive answer. Together with other results, this settles the problem in each dimension.
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We formulate an isomorphic version of the Busemann-Petty problem and solve it in affirmative in the case of sections of proportional dimensions.
We show that the fifth and the eighth Busemann-Petty problems have positive solutions for bodies that are sufficiently close to the Euclidean ball in the Banach-Mazur distance.
We provide an affirmative answer to a variant of the Busemann-Petty problem, proposed by V.~Milman: Let $K$ be a convex body in ${\mathbb R}^n$ and let $D$ be a compact subset of ${\mathbb R}^n$ such that, for some $1\ls k\ls n-1$, $$|P_F(K)|\ls |D\cap F|$$ for all $F\in G_{n,k}$, where $P_F(K)$ is the orthogonal projection of $K$ onto $F$ and $D\cap F$ is the intersection of $D$ with $F$. Then, $$|K|\ls |D|.$$ We also provide estimates for the lower dimensional Busemann-Petty and Shephard problems, and we prove separation in the original Busemann-Petty problem.
We consider the problem of comparing the volumes of two star bodies in an even-dimensional euclidean space $\mathbb R^{2n} = \mathbb C^n$ by comparing their cross sectional areas along complex lines (special 2-dimensional real planes) through the origin. Under mild symmetry conditions on one of the bodies a Busemann-Petty type theorem holds. Quaternionic and Octonionic analogs also hold. The argument relies on integration in polar coordinates coupled with Jensen's inequality. Along the way we provide a criterion that detects which centered bodies are {\it circular}. i.e., stabilized by multipl…
Simultaneous Busemann-Petty and Shephard Volume Comparisons
该研究针对不同维度构造凸体反例，探讨中心超平面截面与正交超平面投影的体积比较能否确定两凸体体积的大小顺序。
H. Busemann and C. M. Petty posed the following problem in 1956: If K and L are origin-symmetric convex bodies in R^n and for each hyperplane H through the origin the volumes of their central slices satisfy vol(K cap H) < vol(L cap H), does it follow that the volumes of the bodies themselves satisfy vol(K) < vol(L)? The problem is trivially positive in R^2. However, a surprising negative answer for n <= 12 was given by Larman and Rogers in 1975. Subsequently, a series of contributions were made to reduce the dimensions to n >= 5 by a number of authors. That is, the problem has a negative answe…
A unified analytic solution to the Busemann-Petty problem was recently found by Gardner, Koldobsky and Schlumprecht. We give an elementary proof of their formulas for the inverse Radon transform.
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in $\C^n$ with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if $n\le 3$ and negative if $n\ge 4.$
We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R^n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R^n and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in R^n such that the ((n-1)-dimensio…
The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in $\mathbb{R}^n$ with smaller volume of all $k$-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to this question is negative if $k>3$. The problem is still open for $k=2,3$. In this article we formulate and completely solve the lower dimensional Busemann-Petty problem in the hyperbolic space $\mathbb{H}^n$.
Busemann-Petty type problems on complex vector spaces
本文研究了复投影、质心和$L_p$-交集体算子的Busemann-Petty型问题，并证明它们与实数情形类似，均可与球面Fourier变换建立联系。
The generalized Busemann-Petty problem asks whether centrally-symmetric convex bodies having larger volume of all m-dimensional sections necessarily have larger volume. When m>3 this is known to be false, but the cases m=2,3 are still open. In those cases, it is shown that when the smaller body's radial function is a (n-m)-th root of the radial function of a convex body, the answer to the generalized Busemann-Petty problem is positive (for any larger star-body). Several immediate corollaries of this observation are also discussed.
The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections. We apply this formula to study properties of general measures most of which were known before only in the case of the standard Lebesgue measure. We solve an analog of the Busemann-Petty problem for the case of general measures. In addition, we show that there are measures, for which the answer to the generalized Busemann-Petty problem is affirmative in all di…
$L_p$ functional Busemann-Petty centroid inequality
本文通过证明 $1 \leq r < n$ 时一类泛函 $r$-混合体积的不等式，建立了 $L_p$ Busemann-Petty 质心不等式的泛函版本。
Sharp affine Sobolev type inequalities via the $\Lp$ Busemann-Petty centroid inequality
本文利用 $\Lp$ Busemann-Petty 质心不等式，为 log-Sobolev、Sobolev 和 Gagliardo-Nirenberg 等尖锐仿射泛函不等式提供了统一且有力的证明方法，并可直接刻画等号成立情形。
A comparison problem for volumes of convex bodies asks whether inequalities $f_K(ξ)\le f_L(ξ)$ for all $ξ\in S^{n-1}$ imply that $\vol_n(K)\le \vol_n(L),$ where $K,L$ are convex bodies in $\R^n,$ and $f_K$ is a certain geometric characteristic of $K.$ By linear stability in comparison problems we mean that there exists a constant $c$ such that for every $\e>0$, the inequalities $f_K(ξ)\le f_L(ξ)+\e$ for all $ξ\in S^{n-1}$ imply that $(\vol_n(K))^{\frac{n-1}n}\le (\vol_n(L))^{\frac{n-1}n}+c\e.$ We prove such results in the settings of the Busemann-Petty and Shephard problems and their generaliz…
A Convex Body Associated to the Busemann Random Simplex Inequality and the Petty conjecture
本文定义凸体$N_p L$，证明其极体的等周不等式等价于$L_p$-Busemann随机单纯形不等式，并由此给出函数版本简单证明、对偶理论及Petty猜想与$L_1$-Sharp仿射Sobolev不等式的等价性。
An affirmative solution to the generalized Busemann--Petty problem with subspace dimensions $2$ and $3$
本文解决了n≥5时m=2、3的广义Busemann-Petty问题低维遗留情形，结合已有结果完成了该问题的完全解答。
We prove the following theorem. Let $μ$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $μ(K\cap H)\le μ(L\cap H)$ for any central hyperplane H. Then $μ(K)\le \sqrt{n} μ(L).$ We also prove this result with better constants for some special classes of measures and bodies. Finally, we prove a version of the hyperplane inequality for convex measures.
Extensions of the Busemann-Petty Problem for Arbitrary Measures
本文扩展了Busemann-Petty问题，研究任意测度下凸体截面与体积的关系，提出更广泛的分布条件并给出实例说明。
We prove stability in the affirmative part of the Busemann-Petty problem on sections of complex convex bodies.
We generalize the hyperplane inequality in dimensions up to 4 to the setting of arbitrary measures in place of the volume. To prove this generalization we establish stability in the affirmative part of the solution to the Busemann-Petty problem asking whether symmetric convex bodies in R^n with smaller (n-1)-dimensional volume of all central hyperplane sections necessarily have smaller n-dimensional volume.
The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions - one geometric, the other probabilistic in nature - are given and shown to lead to the same objects. The geometric approach is then used to establish a number of basic properties of spherical centroid bodies, while the probabilistic approach inspires the proof of a spherical analogue of the classical polar Busemann-Petty centroid inequality.
We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an arbitrary centrally-symmetric convex body in R^n to the class of unit balls of n-dimensional subspaces of L_p-spaces. Finally, we prove a result of the Busemann-Petty type for these moments.
We study the structures of two types of generalizations of intersection-bodies and the problem of whether they are in fact equivalent. Intersection-bodies were introduced by Lutwak and played a key role in the solution of the Busemann-Petty problem. A natural geometric generalization of this problem considered by Zhang, led him to introduce one type of generalized intersection-bodies. A second type was introduced by Koldobsky, who studied a different analytic generalization of this problem. Koldobsky also studied the connection between these two types of bodies, and noted that an equivalence b…
We prove that the (n-2)-dimensional surface area (perimeter) of central hyperplane sections of the n-dimensional unit cube is maximal for the hyperplane perpendicular to the vector (1,1,0,...,0). This gives a positive answer to a question of Pelczynski who solved the three dimensional case. We study both the real and the complex versions of this problem. We also use our result to show that the answer to an analogue of the Busemann-Petty problem for the surface area is negative in dimensions 14 and higher.
Centroids and equilibrium points of convex bodies
本文综述凸体形心与静平衡点相关几何问题的研究成果，涵盖两类形心不等式相关结果、凸体的平衡点分类、平衡点的位置结构与数量及凸多面体静平衡性质等内容。
Several years ago the authors started looking at some problems of convex geometry from a more general point of view, replacing volume by an arbitrary measure. This approach led to new general properties of the Radon transform on convex bodies including an extension of the Busemann-Petty problem and a slicing inequality for arbitrary functions. The latter means that the sup-norm of the Radon transform of any probability density on a convex body of volume one is bounded from below by a positive constant depending only on the dimension. In this note, we prove an inequality that serves as an umbre…
We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $α_n$ depending (or not) on the dimension $n$ so that $$S(K)\leqα_n|K|^{\frac{1}{n}}\max_{ξ\in S^{n-1}}S(K\capξ^{\perp })$$ where $S$ denotes surface area and $|\cdot |$ denotes volume. For any fixed dimension we provide a negative answer to this question, as well as to a weaker version in which sections are replaced by projections onto hyperplanes. We also study the same problem for sections and projections of lower dimension and for all the quermassintegrals of a conve…
This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different volumetric results are considered in a more general setting, with volume replaced by an arbitrary measure. Surprisingly, such a general approach works for a number of volumetric results. In particular, we discuss the Busemann-Petty problem on sections of convex bodies for arbitrary measures and the slicing problem for arbitrary measures. We present generalizations of…
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…
The isomorphic section-projection problem for convex bodies
本文研究了凸体的同构截面-投影问题，证明了在特定条件下凸体体积的关系，并补充了同构Busemann-Petty和Shephard问题。
In this manuscript, we study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if convex bodies $K, L$ satisfy $|K|θ^{\perp}| \le |L \cap θ^{\perp}|$ for all $θ\in S^{n-1}$, then $|K| \le |L|$. Firstly, we study the reverse question: in particular, we show that if $K, L$ are origin-symmetric convex bodies in John's position with $|K \cap θ^{\perp}| \le |L|θ^{\perp}|$ for all $θ\in S^{n-1}$ then $|K| \le \sqrt{n}|L|$. The condition we consider is weaker than both the conditions $|K \cap θ^…
Complex $L_p$-Intersection Bodies
本文引入了范围扩展至$p>-2$的复$L_p$-交体，证明了$\mathbb{S}^1$-不变凸体在不同$p$值下的伪凸性与凸性，并推导出Busemann--Petty型交不等式。
Higher-Order Lp Isoperimetric and Sobolev Inequalities
本研究将凸几何中的维数间算子推广至$L^p$情形，把单纯形拓展为任意含原点的m维凸体，建立了多类m阶$L^p$等周与仿射Sobolev不等式，并给出了线性泛函算子范数体积的等周不等式应用。
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