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)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n-2)-nd derivative of the parallel section functions. The affirmative answer to the Busemann-Petty problem for n\le 4 and the negative answer for n\ge 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.
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.
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…
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 $n\ge 4.$
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 $\mathbb{H}^n$.
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…
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…
An affirmative solution to the generalized Busemann--Petty problem with subspace dimensions $2$ and $3$
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, 200092, China（同济大学数学科学学院，智能计算与应用教育部重点实验室）
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
Universidad Torcuato Di Tella（托尔夸托·迪泰拉大学）；IMAS–CONICET（阿根廷国家科学研究委员会马蒂亚斯·莫林纳物理数学研究所）；Universidad de Sevilla（塞维利亚大学）；Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires（布宜诺斯艾利斯大学理学院）
In this paper, we propose and study the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure $μ$ and a continuous function $φ:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body $K\in\mathcal{K}_0$ such that $K$ is an optimizer of the following optimization problems: \begin{equation*} \inf/\sup \bigg\{\int_{S^{n-1}}φ\big( h_L \big) \,d μ: L \in \mathcal{K}_{0} \ \text{and}\ |L^\circ|=ω_{n}\bigg\}. \end{equation*} The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on $φ,$ …
Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such Minkowski valuations and a generalization of the notion of centroid bodies. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of volume inequalities for inte…
This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, under certain conditions, on the set of convex bodies in terms of the Hausdorff distance. Our proofs rely on the existence of the Orlicz-Petty bodies and the uniform boundedness of the Orlicz-Petty bodies of a convergent sequence of convex bodies. Similar results for the nonhomog…
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.
Busemann's intersection inequality gives an upper bound for the volume of the intersection body of a star body in terms of the volume of the body itself. Koldobsky, Paouris, and Zymonopoulou asked if there is a similar result for $k$-intersection bodies. We solve this problem for star bodies that are close to the Euclidean ball in the Banach-Mazur distance. We also improve a bound obtained by Koldobsky, Paouris, and Zymonopoulou for general star bodies in the case when $k$ is proportional to the dimension.
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.
We prove that if a convex body has absolutely continuous surface area measure, whose density is sufficiently close to the constant, then the sequence $\{Π^mK\}$ of convex bodies converges to the ball with respect to the Banach-Mazur distance, as $m\rightarrow\infty$. Here, $Π$ denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty and to improve a related inequality of Lutwak.
One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior ne…
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…
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…
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…
We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $μ$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k <n.$ If $K$ is a generalized $k$-intersection body, then $$μ(K)\,\leq\,\frac{n}{n-k}c_{n,k}\max_{H} μ(K\cap H) \vol_n(K)^{k/n}.$$ Here $c_{n,k} = |B_2^n|^{(n-k)/n}/|B_2^{n-k}|<1,$ $|B_2^n|$ is the volume of the unit Euclidean ball, and maximum is taken over all $(n-k)$-dimensional subspaces of $\R^n.$ The constant is optimal, and for each intersection body the inequality holds for …
The aim of this note is to investigate the properties of the convex hull and the homothetic convex hull functions of a convex body $K$ in Euclidean $n$-space, defined as the volume of the union of $K$ and one of its translates, and the volume of $K$ and a translate of a homothetic copy of $K$, respectively, as functions of the translation vector. In particular, we prove that the convex hull function of the body $K$ does not determine $K$. Furthermore, we prove the equivalence of the polar projection body problem raised by Petty, and a conjecture of G.Horváth and Lángi about translative constan…
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…
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^…
We prove results relative to the problem of finding sharp bounds for the affine invariant $P(K)=V(ΠK)/V^{d-1}(K)$. Namely, we prove that if $K$ is a 3-dimensional zonoid of volume 1, then its second projection body $Π^2K$ is contained in 8K, while if $K$ is any symmetric 3-dimensional convex body of volume 1, then $Π^2K$ contains 6K. Both inclusions are sharp. Consequences of these results include a stronger version of a reverse isoperimetric inequality for 3-dimensional zonoids-established by the author in a previous work, a reduction for the 3-dimensional Petty conjecture to another isoperim…
This paper gives a systematic study to the general dual-polar Orlicz-Minkowski problem (e.g., Problem \ref{general-dual-polar}). This problem involves the general dual volume $\widetilde{V}_G(\cdot)$ recently proposed in \cite{GHWXY, GHXY} in order to study the general dual Orlicz-Minkowski problem. As $\widetilde{V}_G(\cdot)$ extends the volume and the $q$th dual volume, the general dual-polar Orlicz-Minkowski problem is "polar" to the recently initiated general dual Orlicz-Minkowski problem in \cite{GHWXY, GHXY} and "dual" to the newly proposed polar Orlicz-Minkowski problem in \cite{LuoYeZh…
We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Prékopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geom…
