Franck Barthe, Matthieu Fradelizi, Bernard Maurey
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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。
We formulate an isomorphic version of the Busemann-Petty problem and solve it in affirmative in the case of sections of proportional dimensions.
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.
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 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.
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.
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 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…
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…
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。
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…
We prove stability in the affirmative part of the Busemann-Petty problem on sections of complex convex bodies.
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…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。