Rubin Boris
The generalized Busemann-Petty problem asks whether origin-symmetric convex bodies with lower-dimensional smaller sections necessarily have smaller volume. We study the weighted version of this problem corresponding to the physical situation when bodies are endowed with mass distribution and the relevant sections are measured with attenuation.
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The paper studies possible functional analogs of classical problems from convex geometry. In particular, we provide some bounds in the functional Shephard, Busemann-Petty, and Milman problems generalizing known bounds in this problems for convex sets.
The Busemann-Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ with smaller central hyperplane sections necessarily have smaller $n$-dimensional volume. It is known that the answer is affirmative if $n\le 4$ and negative if $n\ge 5$. In this article we modify the assumptions of the original Busemann-Petty problem to guarantee the affirmative answer in all dimensions.
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in C^n with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n\leq 3 and negative if n\geq 4. In this article we show that the answer remains the same if the volume is replaced by an "almost" arbitrary measure. This result is the complex analogue of Zvavitch's generalization to arbitrary measures of the original real Busemann-Petty problem.
The Busemann-Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ with smaller central hyperplane sections necessarily have smaller $n$-dimensional volume. It is known that the answer to this problem is affirmative if $n\le 4$ and negative if $n\ge 5$. We study this problem in hyperbolic and spherical spaces.
The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space $\mathbb{R}^n$ with smaller central hyperplane sections necessarily have smaller volume. The solution has been completed and the answer is affirmative if $n \le 4$ and negative if $n\ge 5$. In this paper, we investigate the Busemann-Petty problem on entropy of log-concave functions: For even log-concave functions $f$ and $g$ with finite positive integrals in $\mathbb{R}^n$, if the marginal $\int_{\mathbb{R}^n\cap H}f(x)dx$ of $f$ is smaller than the marginal $\int_{\mathbb{R}^n\cap H}g(x)dx$ of $g$ for every…
The Busemann-Petty problem asks whether origin symmetric convex bodies in $\R^n$ with smaller hyperplane sections necessarily have smaller volume. The answer is affirmative if $n\leq 3$ and negative if $n\geq 4.$ We consider a class of convex bodies that have a certain invariance property with respect to their ordered k-tuples of coordinates in $\R^{kn}$ and prove the corresponding problem.
Since the answer to the complex Busemann-Petty problem is negative in most dimensions, it is natural to ask what conditions on the (n-1)-dimensional volumes of the central sections of complex convex bodies with complex hyperplanes allow to compare the n-dimensional volumes. In this article we give necessary conditions on the section function in order to obtain an affirmative answer in all dimensions.
The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. The paper contains a complete solution to the problem when the body with smaller sections is invariant under rotations, preserving mutually orthogonal coordinate subspaces of fixed dimension. The argument relies on the notion of canonical angles between subspaces, spherical Radon transforms, properties of intersection bodies, and the generalized cosine transforms.
The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in R^n with smaller volume of all k-dimensional sections necessarily have smaller volume. The answer is negative for k>3. The problem is still open for k=2,3. We study this problem in the complex hyperbolic n-space and prove that the answer is affirmative only for sections of complex dimension one and negative for sections of higher dimensions.
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.
A Busemann-Petty Type Problem for Dual Radon Transforms
本文研究对偶Radon变换的Busemann-Petty型比较问题，刻画了$p\geq 1$时的解，并证明了反向$L^p$-$L^q$估计，特别反转了Solmon在1979年提出的不等式。
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 $n\ge 4.$
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。
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.
In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty.$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$ in the class of smooth, origin-symmetric convex bodies. First, we prove that the $\infty$-flow evolves suitably normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo $SL(2).$ Second, using the $\infty$-flow and a Harnack estimate for this flow, we prove a stability version of the pl…
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。
Isomorphic Busemann--Petty for arbitrary measures: the sharp order
该研究针对任意测度的同构Busemann-Petty问题，证明了最优常数$C_n$的阶为$\boldsymbol{\boldsymbol{\rho}}$，通过结合前期构造与随机舍入等方法得到了匹配的上下界。
The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in $\mathbb {R}^n$ with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if $n\le 4$ and negative if $n>4$. The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified exposition of this circle of problems in real, complex, and quaternionic $n$-dimensional spaces. All cases are treated simultaneously. In particular, we show that the Busemann-Petty prob…
We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann's theorem to the complex case by proving that complex intersection bodies of symmetric complex convex bodies are also convex. Other results include stability in the complex Busemann-Petty problem for arbitrary measures and the corresponding hyperplane inequality for measures of complex intersection bodies.
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.
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 new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We also prove a sharp upper estimate for the outer volume ratio distance from an arbitrary convex body to the unit balls of subspaces of $L_p$.
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…
Comparison Problems for Radon Transforms
本文研究经典与球面Radon变换的比较问题，刻画使L^p范数比较成立的函数类，推广Lutwak交截面体类并建立相关切片不等式。
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