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1.
Assume that M is a convex body with C2 boundary in d. The paper considers polytopal approximation of M with respect to the most commonly used metrics, like the symmetric difference metric δS, the Lp metric, 1p∞, or the Banach–Mazur metric. In case of δS, the main result states that if Pn is a polytope whose number of k faces is at most n then
The analogous estimates are proved for all the other metrics. Finally, the optimality of these estimates is verified up to a constant depending on the metric and the dimension.  相似文献   

2.
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.  相似文献   

3.
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   

4.
Let F= {C1,C2,...,C} be a family of ndisjoint convex bodies in the plane. We say that a set Vof exterior light sources illuminates F, if for every boundary point of any member of Fthere is a point in Vsuch that is visible from ,i.e. the open line segment joining and is disjoint from F. An illumination system Vis called primitive if no proper subset of Villuminates F. Let pmax(F) denote the maximum number of points forming a primitive illumination system for F, and letpmax(n) denote the minimum of F) taken over all families Fconsisting of ndisjoint convex bodies in the plane. The aim of this paper is to investigate the quantities pmax(F) and pmax(n).  相似文献   

5.
The modified method of refined bounds is proposed and experimentally studied. This method is designed to iteratively approximate convex multidimensional polytopes with a large number of vertices. Approximation is realized by a sequence of convex polytopes with a relatively small but gradually increasing number of vertices. The results of an experimental comparison between the modified and the original methods of refined bounds are presented. The latter was designed for the polyhedral approximation of multidimensional convex compact bodies of general type.  相似文献   

6.
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   

7.
Helly-type results are established relating to the existence of a line supporting a family of nonoverlapping convex bodies in the plane.  相似文献   

8.
The extrapolation design problem for polynomial regression model on the design space [–1,1] is considered when the degree of the underlying polynomial model is with uncertainty. We investigate compound optimal extrapolation designs with two specific polynomial models, that is those with degrees |m, 2m}. We prove that to extrapolate at a point z, |z| > 1, the optimal convex combination of the two optimal extrapolation designs | m * (z), 2m * (z)} for each model separately is a compound optimal extrapolation design to extrapolate at z. The results are applied to find the compound optimal discriminating designs for the two polynomial models with degree |m, 2m}, i.e., discriminating models by estimating the highest coefficient in each model. Finally, the relations between the compound optimal extrapolation design problem and certain nonlinear extremal problems for polynomials are worked out. It is shown that the solution of the compound optimal extrapolation design problem can be obtained by maximizing a (weighted) sum of two squared polynomials with degree m and 2m evaluated at the point z, |z| > 1, subject to the restriction that the sup-norm of the sum of squared polynomials is bounded.  相似文献   

9.
It is proven that if Q is convex and w(x)= exp(-Q(x)) is the corresponding weight, then every continuous function that vanishes outside the support of the extremal measure associated with w can be uniformly approximated by weighted polynomials of the form w n P n . This solves a problem of P. Borwein and E. B. Saff. Actually, a similar result is true locally for any parts of the extremal support where Q is convex. February 10, 1998. Date revised: July 23, 1998. Date accepted: August 17, 1998.  相似文献   

10.
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.  相似文献   

11.
In his book “Geometric Tomography” Richard Gardner asks the following question. Let P and Q be origin-symmetric convex bodies in R3 whose sections by any plane through the origin have equal perimeters. Is it true that P=Q? We show that the answer is “Yes” in the class of origin-symmetric convex polytopes. The problem is treated in the general case of Rn.  相似文献   

12.
For a compact set K\subset R d with nonempty interior, the Markov constants M n (K) can be defined as the maximal possible absolute value attained on K by the gradient vector of an n -degree polynomial p with maximum norm 1 on K . It is known that for convex, symmetric bodies M n (K) = n 2 /r(K) , where r(K) is the ``half-width' (i.e., the radius of the maximal inscribed ball) of the body K . We study extremal polynomials of this Markov inequality, and show that they are essentially unique if and only if K has a certain geometric property, called flatness. For example, for the unit ball B d (\smallbf 0, 1) we do not have uniqueness, while for the unit cube [-1,1] d the extremal polynomials are essentially unique. September 9, 1999. Date revised: September 28, 2000. Date accepted: November 14, 2000.  相似文献   

13.
Categorical data of high (but finite) dimensionality generate sparsely populated J-way contingency tables because of finite sample sizes. A model representing such data by a "smooth" low dimensional parametric structure using a "natural" metric would be useful. We discuss a model using a metric determined by convex sets to represent moments of a discrete distribution to order J. The model is shown, from theorems on convex polytopes, to depend only on the linear space spanned by the convex set—it is otherwise measure invariant. We provide an empirical example to illustrate the maximum likelihood estimation of parameters of a particular statistical application (Grade of Membership analysis) of such a model.  相似文献   

14.
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.  相似文献   

15.
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   

16.
17.
Pointwise estimates are obtained for simultaneous approximation of a function f and its derivatives by means of an arbitrary sequence of bounded projection operators with some extra condition (1.3) (we do not require the operators to be linear) which map C[-1,1] into polynomials of degree n, augmented by the interpolation of f at some points near ±1. The present result essentially improved those in [BaKi3], and several applications are discussed in Section 4.  相似文献   

18.
Let K be a closed bounded convex subset of R n ; then by a result of the first author, which extends a classical theorem of Whitney there is a constant w m (K) so that for every continuous function f on K there is a polynomial ϕ of degree at most m-1 so that |f(x)-ϕ(x)|≤ w_m(K) sup _{x,x+mh∈ K} |Δ_h^m(f;x)|. The aim of this paper is to study the constant w m (K) in terms of the dimension n and the geometry of K . For example, we show that w 2 (K)≤ (1/2) [ log 2 n]+5/4 and that for suitable K this bound is almost attained. We place special emphasis on the case when K is symmetric and so can be identified as the unit ball of finite-dimensional Banach space; then there are connections between the behavior of w m (K) and the geometry (particularly the Rademacher type) of the underlying Banach space. It is shown, for example, that if K is an ellipsoid then w 2 (K) is bounded, independent of dimension, and w 3 (K)\sim log n . We also give estimates for w 2 and w 3 for the unit ball of the spaces l p n where 1≤ p≤∈fty. September 24, 1997. Dates revised: January 18, 1999 and June 10, 1999. Date accepted: June 25, 1999.  相似文献   

19.
最近.许树声[1]撰文给出了对L_p范数下凸约束最佳逼近的特征定理.并研究了该定理的苦干应用.然而,文[1]引理1证明中部分地方有误.本文给予了纠正,予以重新证明  相似文献   

20.
In this paper we consider the two events that a random congruent copy of a convex body meets each one of two given families of equidistant lines in the plane. The probabilities are easily calculated. Then it is discovered that there always exists a value for the angle between the nonparallel lines, such that the two events be independent. For convex bodies of constant width, and only for them, the two events are independent for any .  相似文献   

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