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.     

Intersections of Convex Bodies

Let be nonempty convex bodies in . Let be vectors in , let , and let . Then is a convex set, and the family of sets is concave. Let . Then for the mean cross-sectional measures W_v (\Phi (\rho )), , the functions are concave on D. (Note that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaca% WGxbWaaSbaaSqaaiaaicdaaeqaaOGaaiikaiabfA6agjaacIcacqaH% bpGCcaGGPaGaaiykaiabg2da9iaabAfacaqGVbGaaeiBamaaBaaale% aatCvAUfKttLearyqr1ngBPrgaiuGacqWFRbWAaeqaaOGaeuOPdyKa% aiikaiabeg8aYjaacMcaaaa!4EE7!\[W_0 (\Phi (\rho )) = {\text{Vol}}_k\Phi (\rho )\] is the k-volume.) Bibliography: 2 titles.     

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.

     

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．     

Mixed Functionals of Convex Bodies

We define a class of real functions on tuples of convex bodies. They are a common generalization of mixed volumes and of certain functionals which have been studied in translative integral geometry. For polytopes, these functionals have various explicit representations in terms of volumes of lower-dimensional faces. For the mentioned functionals from integral geometry, these representations generalize a result of Weil and answer a question posed by Janson. Received January 5, 1999, and in revised form March 12, 1999. Online publication May 16, 2000.     

Convex Bodies with Homothetic Sections

We prove that if K is a convex body in Eⁿ⁺¹, n2, and p₀ is apoint of K with the property that all n-sections of K throughp₀ are homothetic, then K is a Euclidean ball.     

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 of Convex Bodies by ConvexBodies

For the affine distance d(C,D) between two convex bodies C, D(?) Rn, 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) < n1/2 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) < n2 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 asymmetry for convex bodies.     

The Blocking Numbers of Convex Bodies

Besides determining the exact blocking numbers of cubes and balls, a conditional lower bound for the blocking numbers of convex bodies is achieved. In addition, several open problems are proposed. Received December 11, 1998, and in revised form October 5, 1999.     

Directed Projection Functions of Convex Bodies

For 1 ≤ i < j < d, a j-dimensional subspace L of and a convex body K in , we consider the projection K|L of K onto L. The directed projection function v _i,j (K;L,u) is defined to be the i-dimensional size of the part of K|L which is illuminated in direction u ∈ L. This involves the i-th surface area measure of K|L and is motivated by Groemer’s [17] notion of semi-girth of bodies in . It is well-known that centrally symmetric bodies are determined (up to translation) by their projection functions, we extend this by showing that an arbitrary body is determined by any one of its directed projection functions. We also obtain a corresponding stability result. Groemer [17] addressed the case i = 1, j = 2, d = 3. For j > 1, we then consider the average of v _1,j (K;L,u) over all spaces L containing u and investigate whether the resulting function determines K. We will find pairs (d,j) for which this is the case and some pairs for which it is false. The latter situation will be seen to be related to some classical results from number theory. We will also consider more general averages for the case of centrally symmetric bodies. The research of the first author was supported in part by NSF Grant DMS-9971202 and that of the second author by a grant from the Volkswagen Foundation.     

On Extensive Subsets of Convex Bodies

A subset S of a d-dimensional convex body K is extensive if S ∂K and for any p, q ∈ S the distance between p and q is at least one-half of the maximum length of chords of K parallel to the segment pq. In this paper we establish the general upper bound |S| ≤ 3 ^d — 1. We also find an upper bound for a certain class of 3-polytopes, which leads to the determination of the maximum cardinalities of extensive subsets and their extremal configurations for tetrahedra, octahedra and some other 3-polytopes. This revised version was published online in June 2006 with corrections to the Cover Date.     

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．     

Convex Bodies in Exceptional Relative Positions

Two convex bodies K and K' in Euclidean space Eⁿ can be saidto be in exceptional relative position if they have a commonboundary point at which the linear hulls of their normal coneshave a non-trivial intersection. It is proved that the set ofrigid motions g for which K and gK' are in exceptional relativeposition is of Haar measure zero. A similar result holds trueif ‘exceptional relative position’ is defined viacommon supporting hyperplanes. Both results were conjecturedby S. Glasauer; they have applications in integral geometry.     

Lattice Points in Large Convex Bodies

 Denote by the number of points of the lattice in the “blown up” domain , where is a convex body in () whose boundary is smooth and has nonzero curvature throughout. It is proved that for every fixed

On the Distance between Convex Bodies

ForanytwoconvexbodiesCandDlntheEuc1ideanspaceR"denotebyd(C,D)theminimalnumberAsuchthatthereexistsanaffineoperatorAwhichsatisfiesA(C)=D=AA(C),whereAA(C)denotesahomotheticcopyofA(C)withhom0thetycentrelnsomepointofA(C).d(C,D)iscalledtheMinkowskid1stancebetweenCandD.JohnL'jprovedthatifCisanellipsoid,thend(C,D)相似文献   

Lattice Points in Large Convex Bodies

 Denote by the number of points of the lattice in the “blown up” domain , where is a convex body in () whose boundary is smooth and has nonzero curvature throughout. It is proved that for every fixed

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.