Approximation of Convex Bodies by Centrally Symmetric Bodies

We present an analog of the well-known theorem of F. John about the ellipsoid of maximal volume contained in a convex body. Let C be a convex body and let D be a centrally symmetric convex body in the Euclidean d-space. We prove that if D is an affine image of D of maximal possible volume contained in C, then C a subset of the homothetic copy of D with the ratio 2d-1 and the homothety center in the center of D. The ratio 2d-1 cannot be lessened as a simple example shows.     

凸体的包含测度 (英)

凸体的包含测度理论是积分几何中一个十分重要的课题,它提供了一种研究凸体性质的全新方法．本文获得了关于凸体包含测度的几个不等式,从而证明了包含测度对于凸体没有线性．     

Stability of the Minkowski Measure of Asymmetry for Convex Bodies

Given a convex body $C\subset R^n$ (i.e., a compact convex set with nonempty interior), for $x\in$ {\it int}$(C)$, the interior, and a hyperplane $H$ with $x\in H$, let $H_1,H_2$ be the two support hyperplanes of $C$ parallel to $H$. Let $r(H, x)$ be the ratio, not less than 1, in which $H$ divides the distance between $H_1,H_2$. Then the quantity $${\it As}(C):=\inf_{x\in {\it int}(C)}\,\sup_{H\ni x}\,r(H,x)$$ is called the Minkowski measure of asymmetry of $C$. {\it As}$(\cdot)$ can be viewed as a real-valued function defined on the family of all convex bodies in $R^n$. It has been known for a long time that {\it As}$(\cdot)$ attains its minimum value 1 at all centrally symmetric convex bodies and maximum value $n$ at all simplexes. In this paper we discuss the stability of the Minkowski measure of asymmetry for convex bodies. We give an estimate for the deviation of a convex body from a simplex if the corresponding Minkowski measure of asymmetry is close to its maximum value. More precisely, the following result is obtained: Let $C\subset R^n$ be a convex body. If {\it As}$(C)\ge n-\varepsilon$ for some $0\le \varepsilon < 1/8(n+1),$ then there exists a simplex $S_0$ formed by $n+1$ support hyperplanes of $C$, such that $$(1+8(n+1)\varepsilon)^{-1}S_0\subset C\subset S_0,$$ where the homethety center is the (unique) Minkowski critical point of $C$. So $$d_{{\rm BM}}(C,S)\le 1+8(n+1)\varepsilon$$ holds for all simplexes $S$, where $d_{{\rm BM}}(\cdot,\cdot)$ denotes the Banach-Mazur distance.     

Covering Convex Bodies by Cylinders and Lattice Points by Flats

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 1≤k≤d−1. K. Bezdek and A.E. Litvak are partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.     

On Prop erties of p-critical Points of Convex Bo dies

Properties of the p-measures of asymmetry and the corresponding affine equivariant p-critical points, defined recently by the second author, for convex bodies are discussed in this article. In particular, the continuity of p-critical points with respect to p on(1, +∞) is confirmed, and the connections between general p-critical points and the Minkowski-critical points(∞-critical points) are investigated. The behavior of p-critical points of convex bodies approximating a convex bodies is studied as well.     

凸体间的逼近

For the affine distance d(C, D) between two convex bodies C, D C R^n, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C, D) ≤ n^1/2 if one is an ellipsoid and another is symmetric, d(C, D) ≤ n if both are symmetric, and from F. John's result and d(C1, C2) ≤ d(C1, C3)d(C2, C3) one has d(C, D) ≤ n^2 for general convex bodies; M. Lassak proved d(C, D) ≤ (2n - 1) if one of them is symmetric. In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asvmmetrv for convex bodies.     

Measure-Valued Valuations and Mixed Curvature Measures of Convex Bodies

We prove a polynomial expansion for measure-valued functionals which are translation covariant on the set of convex bodies. The coefficients are measures on product spaces. We then apply this construction to the curvature measures of convex bodies and obtain mixed curvature measures for bodies in general relative position. These are used to generalize an integral geometric formula for nonintersecting convex bodies. Finally, we introduce support measures relative to a quite general structuring body B and describe connections between the different types of measures.     

凸体的p次径向平均体的几个性质

本文证明了凸体的p次径向平均体的线性不变性和椭球的p次径向平均体仍是椭球,并且获得了p次径向平均体相应于Blaschke-Santaló不等式的一个反向不等式．     

凸体间几种仿射不变距离的等价性与估计

本文研究了凸体间(绝对)Banach-Mazur距离各种不同的定义,证明了它们的等价性;给出了Banach-Mazur距离与绝对Banach-Mazur距离相等的一个充分条件;最后研究了凸体极体间的BanachMazur距离,并对特殊凸体对证明了其Banach-Mazur距离与其某一对极体间的Banach-Mazur距离相等.文中结果为Banach-Mazur距离最佳上界的估计提供了进一步研究的基础.     

On Convex Class of Pairs of Convex Bodies

In this paper we introduce a quotient class of pairs of convex bodies in which every member have convex union.

     

The Cross-Section Body, Plane Sections of Convex Bodies and Approximation of Convex Bodies, II

We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, those of sections of convex bodies and of sections of their circumscribed cylinders. For L ^d a convex body, we take n random segments in L and consider their 'Minkowski average' D. For fixed n, the pth moments of V(D) (1 p < ) are minimized, for V (L) fixed, by the ellipsoids. For k = 2 and fixed n, the pth moment of V(D) is maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.     

Approximation to Probabilities Through Uniform Laws on Convex Sets

Let P be a probability distribution on ^d and let be the family of the uniform probabilities defined on compact convex sets of ^d with interior non-empty. We prove that there exists a best approximation to P in , based on the L ₂-Wasserstein distance. The approximation can be considered as the best representation of P by a convex set in the minimum squares setting, improving on other existent representations for the shape of a distribution. As a by-product we obtain properties related to the limit behavior and marginals of uniform distributions on convex sets which can be of independent interest.     

Primitive Inner Illuminating Systems for Convex Bodies

A set X of boundary points of a (possibly unbounded) convex body KE ^d illuminating K from within is called primitive if no proper subset of X still illuminates K from within. We prove that for such a primitive set X of an unbounded, convex set KE ^d (distinct from a cone) one has X=2 if d=2, X6 if d=3, and that there is no upper bound for X if d4.     

On Reverses of the Blaschke-Santalo Inequality for Convex Bodies

Reisner proved a reverse of the Blaschke-Santal5 inequality for zonoid bodies, Bourgain and Milman showed another reverse of the Blaschke-Santal5 inequality for centered convex bodies. In this paper, two reverses of the Blaschke-Santal5 inequality for convex bodies are given by the Petty projection inequality and above two reverses. Further, using above methods, we also obtain two analogues of the Petty＇s conjecture for projection bodies, respectively.     

Determination of Convex Bodies by Projection Functions

This paper is concerned with an investigation to what extent a convex body is determined by projection functions. We introduce the notion of convex bodies that are almost determined by their brightness function and give a geometric characterization of these bodies. We further prove that, for i 2,...,d–2, a polytope is determined by its ith projection function, if all of its (i+1)-faces are parallelepipeds. Both results extend work by Gardner and Voli [4].     

Convex Quadratic Approximation

For some applications it is desired to approximate a set of m data points in ⁿ with a convex quadratic function. Furthermore, it is required that the convex quadratic approximation underestimate all m of the data points. It is shown here how to formulate and solve this problem using a convex quadratic function with s = (n + 1)(n + 2)/2 parameters, s m, so as to minimize the approximation error in the L ¹ norm. The approximating function is q(p,x), where p ^s is the vector of parameters, and x n. The Hessian of q(p,x) with respect to x (for fixed p) is positive semi-definite, and its Hessian with respect to p (for fixed x) is shown to be positive semi-definite and of rank n. An algorithm is described for computing an optimal p* for any specified set of m data points, and computational results (for n = 4,6,10,15) are presented showing that the optimal q(p*,x) can be obtained efficiently. It is shown that the approximation will usually interpolate s of the m data points.     

Universal Polynomial Majorants on Convex Bodies

Let K be a convex body in ^d (d2), and denote by B_n(K) the set of all polynomials p_n in ^d of total degree n such that |p_n|1 on K. In this paper we consider the following question: does there exist a p*_nB_n(K) which majorates every element of B_n(K) outside of K? In other words can we find a minimal γ1 and p*_nB_n(K) so that |p_n(x)|γ |p*_n(x)| for every p_nB_n(K) and x ^d\K? We discuss the magnitude of γ and construct the universal majorants p*_n for evenn. It is shown that γ can be 1 only on ellipsoids. Moreover, γ=O(1) on polytopes and has at most polynomial growth with respect to n, in general, for every convex body K.     

On Gaussian Marginals of Uniformly Convex Bodies

Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L _p for 1<p<∞. The same is true when L _p is replaced by S _p ^m , the l _p -Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies. Supported in part by BSF and ISF.     

Multiple Integral Inequalities for Schur Convex Functions on Symmetric and Convex Bodies

In this paper, by making use of Divergence theorem for multiple integrals, we establish some integral inequalities for Schur convex functions defined on bodies $B⊂\mathbb{R}^n$ that are symmetric, convex and have nonempty interiors. Examples for three dimensional balls are also provided.     

On the Determination of Convex Bodies by Translates of Their Projections

It is a well-known fact that a three-dimensional convex body is, up to translations, uniquely determined by the translates of its orthogonal projections onto all planes. Simple examples show that this is no longer true if only lateral projections are permitted, that is orthogonal projections onto all planes that contain a given line. In this article large classes of convex bodies are specified that are essentially determined by translates or homothetic images of their lateral projections. The problem is considered for all dimensions , and corresponding stability results are proved. Finally, it is investigated to which degree of precision a convex body can be determined by a finite number of translates of its projections. Various corollaries concern characterizations and corresponding stability statements for convex bodies of constant width and spheres.