共查询到20条相似文献,搜索用时 15 毫秒
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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. 相似文献
3.
V. A. Zalgaller 《Journal of Mathematical Sciences》2001,104(4):1255-1258
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. 相似文献
4.
Jerzy Grzybowski Ryszard Urbanski 《Proceedings of the American Mathematical Society》1997,125(11):3397-3401
In this paper we introduce a quotient class of pairs of convex bodies in which every member have convex union.
5.
Marek Lassak 《Geometriae Dedicata》1998,72(1):63-68
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. 相似文献
6.
We prove a Hadwiger transversal-type result, characterizing convex position on a family of non-crossing convex bodies in the plane. This theorem suggests
a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type turns out to
be an oriented matroid. We also give new upper bounds on the Erdős–Szekeres theorem in the context of convex bodies. 相似文献
7.
8.
R. Schneider 《Discrete and Computational Geometry》2000,24(2-3):527-538
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. 相似文献
9.
10.
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. 相似文献
11.
Geometric Tomography of Convex Cones 总被引:1,自引:0,他引:1
Gabriele Bianchi 《Discrete and Computational Geometry》2009,41(1):61-76
The parallel X-ray of a convex set K⊂ℝ
n
in a direction u is the function that associates to each line l, parallel to u, the length of K∩l. The problem of finding a set of directions such that the corresponding X-rays distinguish any two convex bodies has been widely studied in geometric tomography. In this paper we are interested in
the restriction of this problem to convex cones, and we are motivated by some applications of this case to the covariogram
problem. We prove that the determination of a cone by parallel X-rays is equivalent to the determination of its sections from a different type of tomographic data (namely, point X-rays of a suitable order). We prove some new results for the corresponding problem which imply, for instance, that convex
polyhedral cones in ℝ3 are determined by parallel X-rays in certain sets of two or three directions. The obtained results are optimal. 相似文献
12.
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. 相似文献
13.
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. 相似文献
14.
L. Dalla D. G. Larman P. Mani-Levitska C. Zong 《Discrete and Computational Geometry》2000,24(2-3):267-278
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. 相似文献
15.
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. 相似文献
16.
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. 相似文献
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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. 相似文献
19.
Andrs Kro 《Journal of Approximation Theory》2001,111(2):303
Let K be a convex body in
d (d2), and denote by Bn(K) the set of all polynomials pn in
d of total degree n such that |pn|1 on K. In this paper we consider the following question: does there exist a p*nBn(K) which majorates every element of Bn(K) outside of K? In other words can we find a minimal γ1 and p*nBn(K) so that |pn(x)|γ |p*n(x)| for every pnBn(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. 相似文献
20.
Wolfgang Müller 《Monatshefte für Mathematik》1999,44(2):315-330
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
where for and . This improves a classic result of E. Hlawka [8] and its refinements due to E. Kr?tzel and W. G. Nowak ([14], [15]). The proof uses a multidimensional variant of the method of van der Corput for the estimation of exponential
sums. 相似文献