M. Yaskina
We construct symmetric convex bodies that are not intersection bodies, but all of their central hyperplane sections are intersection bodies. This result extends the studies by Weil in the case of zonoids and by Neyman in the case of subspaces of $L_p$.
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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。
We establish a family of isoperimetric inequalities for sets that interpolate between intersection bodies and dual Lp centroid bodies. This provides a bridge between the Busemann intersection inequality and the Lutwak--Zhang inequality. The approach depends on new empirical versions of these inequalities.
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 study isomorphic properties of two generalizations of intersection bodies, the class of k-intersection bodies and the class of generalized k-intersection bodies. We also show that the Banach-Mazur distance of the k-intersection body of a convex body, when it exists and it is convex, with the Euclidean ball, is bounded by a constant depending only on k, generalizing a well-known result of Hensley and Borell. We conclude by giving some volumetric estimates for k-intersection bodies.
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.
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.
Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. The main focus of this article is interrelation between generalized cosine transforms of different kinds in the context of their application to investigation of a certain family of intersection bodies, which we call $\lam$-intersection bodies. The latter include $k$-intersection bodies (in the sense of A. Koldobsky) and unit balls of finite-dimensional subspaces of $L_p$-spaces. In particul…
In this paper we study how certain symmetries of convex bodies affect their geometric properties. In particular, we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations, generalizing complex and quaternionic convex bodies. We conduct a systematic study of sections of bodies with symmetries of this type, with the emphasis on problems of the Busemann-Petty type and hyperplane inequalities. The main role belongs to the class of intersection bodies with symmetries.
Let $\mathcal I_k$ be the class of convex $k$-intersection bodies in $\mathbb{R}^n$ (in the sense of Koldobsky) and $\mathcal I_k^m$ be the class of convex origin-symmetric bodies all of whose $m$-dimensional central sections are $k$-intersection bodies. We show that 1) $\mathcal I_k^m\not\subset \mathcal I_k^{m+1}$, $k+3\le m<n$, and 2) $\mathcal I_l \not\subset \mathcal I_k$, $1\le k<l < n-3$.
We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in John's position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in John's position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.
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 $n\ge 4.$
In the paper "Isoperimetry of waists and local versus global asymptotic convex geometries", it was proved that the existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of randomly rotated K and L is nicely bounded. In this appendix, we achieve a polynomial bound on the diameter of that intersection (in the ratio of the dimensions of the sections).
Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and $s$-concave measures
In 2000, A. Koldobsky asked whether two types of generalizations of the notion of an intersection-body, are in fact equivalent. The structures of these two types of generalized intersection-bodies have been studied by the author in [http://www.arxiv.org/math.MG/0512058], providing substantial positive evidence for a positive answer to this question. The purpose of this note is to construct a counter-example, which provides a surprising negative answer to this question in a strong sense. This implies the existence of non-trivial non-negative functions in the range of the spherical Radon transfo…
On determination of convex bodies by their section functions
本文研究凸体由平行截面函数的局部信息确定的问题，刻画了局部可分离截面函数的凸体，推进了浮体同伦猜想，并解决了Barker和Larman关于无穷多个距离处截面函数确定性的问题。
We show that if $d\ge 4$ is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.
We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume ratio. We also prove it for arbitrary symmetric convex bodies under the condition that the dimension of sections is less than $λn$ for some $λ\in (0,1).$ The constant depends only on $λ.$ Finally, we show that the behavior of the minimal sections for some measures may be different from the case of volume.
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…
Radon Transforms of Projection Functions and Aleksandrov-Fenchel Inequalities
本文针对平均截面体的对数凹性问题，建立新的亚历山德罗夫-芬切尔不等式，证明平均截面体的所有内在体积均为对数凹的。
On Sections of Convex Bodies in John's Position and of Generalised $B_p^n$ Balls
本文通过结合几何Brascamp-Lieb不等式与广义Parseval型恒等式，改进了K.Ball的论证，为约翰位置下凸体截面体积提供了精确估计，并推广了Meyer-Pajor和Koldobsky关于$B_p^n$球体极值截面的经典估计。
We say that an even continuous function $H$ on the unit sphere $Ω$ in $R^n$ admits the Blaschke-Levy representation with $q>0$ if there exists an even function $b\in L_1(Ω)$ so that $H^q(x)=\int_Ω|(x,ξ)|^q b(ξ)\ dξ$ for every $x\in Ω.$ This representation has numerous applications in convex geometry, probability and Banach space theory. In this paper, we present a simple formula (in terms of the derivatives of $H$) for calculating $b$ out of $H.$ We use this formula to give a sufficient condition for isometric embedding of a space into $L_p$ which contributes to the 1937 P.Levy's problem and t…
We obtain explicit inversion formulas for the Radon-like transform that assigns to a function on the unit sphere the integrals of that function over hemispheres lying in lower dimensional central cross-sections. The results are applied to determination of star bodies from the volumes of their central half-sections.
Dual to Koldobsky's notion of j-intersection bodies, the class of j-projection bodies is introduced, generalizing Minkowski's classical notion of projection bodies of convex bodies. A Fourier analytic characterization of j-projection bodies in terms of their area measures of order j is obtained. In turn, this yields an equivalent characterization of j-projection bodies involving Alesker's Fourier type transform on translation invariant smooth spherical valuations. As applications of these results, several basic properties of j-projection bodies are established and new non-trivial examples are …
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.
Centroids of sections of convex bodies and Lusternik-Schnirelmann category
本文利用Lusternik-Schnirelmann范畴理论，证明了在特定条件下凸体截面的质心属于边界。
Extremal sections and projections of certain convex bodies: a survey
本文综述了$\ell_p$球等凸体在低维子空间上截面与投影体积的极值估计，回顾了源于数的几何的概率与分析方法的进展。
A Rényi entropy interpretation of anti-concentration and noncentral sections of convex bodies
本文通过雷尼熵解释了反集中现象，并给出了各向同性凸体非中心截面体积的精确界。
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.
Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies.
In this paper, we construct two convex bodies $K$ and $L$ in $\mathbb{R}^n$, $n\geq 3$, such that their projections $K|H$, $L|H$ onto every subspace $H$ are congruent, but nevertheless, $K$ and $L$ do not coincide up to a translation or a reflection in the origin. This gives a negative answer to an old conjecture posed by Nakajima and Süss.
We obtain a new upper estimate on the Euclidean diameter of the intersection of the kernel of a random matrix with iid rows with a given convex body. The proof is based on a small-ball argument rather than on concentration and thus the estimate holds for relatively general matrix ensembles.
We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if $K$ is a convex body in ${\mathbb R}^n$ with $0\in {\rm int}(K)$ and if $μ$ is a measure on ${\mathbb R}^n$ with a locally integrable non-negative density $g$ on ${\mathbb R}^n$, then \begin{equation*}μ(K)\leq \left (c\sqrt{n-k}\right )^k\max_{F\in G_{n,n-k}}μ(K\cap F)\cdot |K|^{\frac{k}{n}}\end{equation*} for every $1\leq k\leq n-1$. Also, if $μ$ is even and log-concave, and if $K$ is a sym…
This is a somewhat expanded form of a four hours course given, with small variations, first at the educational workshop Probabilistic methods in Geometry, Bedlewo, Poland, July 6-12, 2008 and a few weeks later at the Summer school on Fourier analytic and probabilistic methods in geometric functional analysis and convexity, Kent, Ohio, August 13-20, 2008. The main part of these notes gives yet another exposition of Dvoretzky's theorem on Euclidean sections of convex bodies with a proof based on Milman's. This material is by now quite standard. Towards the end of these notes we discuss issues re…
Let $K$ be an isotropic symmetric convex body in ${\mathbb R}^n$. We show that a subspace $F\in G_{n,n-k}$ of codimension $k=γn$, where $γ\in (1/\sqrt{n},1)$, satisfies $$K\cap F\subseteq \frac{c}{γ}\sqrt{n}L_K (B_2^n\cap F)$$ with probability greater than $1-\exp (-\sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(μ)$ of an isotropic log-concave probability measure $μ$ on ${\mathbb R}^n$. For every $1\leq q\leq n$ and $γ\in (0,1)$ we show that a random subspace $F\in G_{n,(1-γ)n}$ satisfies $Z_q(μ)\cap F\subseteq c_2(γ)\sqrt{q}\,B_2^n\cap F$. W…
Lower bounds on non-central sections of isotropic convex bodies
针对固定距离 t0，给出对称各向同性凸体 K 的 L_K 乘以超平面截面的体积的渐近锐利下界。
We show that the hyperplane conjecture holds for the classes of $k$-intersection bodies with arbitrary measures in place of volume.
Convex bodies with algebraic section volume functions
本文研究凸体截面体积函数的代数性质，证明在光滑严格凸体中，椭球体可由特定多项式代数方程完全刻画，从而推广了多项式可积区域的相关结论。
This article gives estimates on covering numbers and diameters of random proportional sections and projections of symmetric quasi-convex bodies in $\mathbb R$. These results were known for the convex case and played an essential role in development of the theory. Because duality relations can not be applied in the quasi-convex setting, new ingredients were introduced that give new understanding for the convex case as well.
It was shown by G. Pisier that any finite-dimensional normed space admits an $α$-regular $M$-position, guaranteeing not only regular entropy estimates but moreover regular estimates on the diameters of minimal sections of its unit-ball and its dual. We revisit Pisier's argument and show the existence of a \emph{different} position, which guarantees the same estimates for \emph{randomly sampled} sections \emph{with high-probability}. As an application, we obtain a \emph{random} version of V. Milman's Quotient-of-Subspace Theorem, asserting that in the above position, \emph{typical} quotients of…
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.
It is proved that for a symmetric convex body K in R^n, if for some tau > 0, |K cap (x+tau K)| depends on ||x||_K only, then K is an ellipsoid. As a part of the proof, smoothness properties of convolution bodies ls are studied.
In this paper we compare the different phenomena that occur when intersecting geometric objects with random geodesics on the unit sphere and inside convex bodies. On the high dimensional sphere we see that with probability bounded away from zero, the observed length will deviate from the actual measure by at most a fixed error for any subset, while in convex bodies we can always choose a subset for which the behavior would be close to a zero-one law, as the dimension grows. The result for the sphere is based on an analysis of the Radon transform. Using similar tools we analyze the variance of …
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 find a sharp combinatorial bound for the metric entropy of sets in R^n and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central Limit Theorem if the square root of its combinatorial dimension is integrable. 2. The uniform entropy is equivalent to the combinatorial dimension under minimal regularity. Our method also constructs a nicely bounded coordinate section of a symmetric convex body in R^n. In the operator theory, this essentially proves for all normed spaces the restricted invert…
In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as the Birch theorem) and complex polynomials. We also consider a stronger conjecture on the homogeneous polynomial fields in the canonical bundle over real and complex Grassmannians. This conjecture is much stronger and false in general, but it is proved in the cases of $d=2$ (for $k$'s of certain type), odd $d$, and the complex Grassmannian (for odd and even…
Existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of random rotations of K and L is nicely bounded. For L = subspace, this main result immediately yields the unexpected phenomenon: "If K has one nicely bounded section, then most sections of K are nicely bounded". This 'existence implies randomness' consequence was proved independently in [Giannopoulos, Milman and Tsolomitis]. The main result represents a new connection between the local asymptotic convex geometry (study of sections of convex bodies) and the global asymptotic convex geomet…
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.
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.
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…
On the Fourier Mean Bodies of a Convex Body
本文引入凸体K的傅里叶p次平均体F_pK，通过傅里叶变换连接径向p次平均体R_pK，证明傅里叶反演公式，并研究其仿射几何性质、凸性范围及与相交体的关系。
We prove that for any compact set B in R^d and for any epsilon >0 there is a finite subset X of B of |X|=d^{O(1/epsilon^2)} points such that the maximum absolute value of any linear function ell: R^d --> R on X approximates the maximum absolute value of ell on B within a factor of epsilon sqrt{d}. We also discuss approximations of convex bodies by projections of spectrahedra, that is, by projections of sections of the cone of positive semidefinite matrices by affine subspaces.
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