共查询到20条相似文献,搜索用时 15 毫秒
1.
Classical problems in integral geometry and geometric probability involve the kinematic measure of congruent segments of fixed length within a convex body in R3. We give this measure from rotational formulae; that is, from isotropic plane sections through a fixed point. From this result we also obtain a new rotational formula for the volume of a convex body; which is proved to be equivalent to the wedge formula for the volume. 相似文献
2.
Integral section formulae for totally geodesic submanifolds (planes) intersecting a compact submanifold in a space form are available from appropriate representations of the motion invariant density (measure) of these planes. Here we present a new decomposition of the invariant density of planes in space forms. We apply the new decomposition to rewrite Santaló's sectioning formula and thereby to obtain new mean values for lines meeting a convex body. In particular we extend to space forms a recently published stereological formula valid for isotropic plane sections through a fixed point of a convex body in R3. 相似文献
3.
A.E. Litvak 《Journal of Functional Analysis》2006,231(2):438-457
We study the diameters of sections of convex bodies in RN determined by a random N×n matrix Γ, either as kernels of Γ* or as images of Γ. Entries of Γ are independent random variables satisfying some boundedness conditions, and typical examples are matrices with Gaussian or Bernoulli random variables. We show that if a symmetric convex body K in RN has one well bounded k-codimensional section, then for any m>ck random sections of K of codimension m are also well bounded, where c?1 is an absolute constant. It is noteworthy that in the Gaussian case, when Γ determines randomness in sense of the Haar measure on the Grassmann manifold, we can take c=1. 相似文献
4.
L. Caravenna 《Journal of Functional Analysis》2010,258(11):3604-3661
We consider the disintegration of the Lebesgue measure on the graph of a convex function f:Rn→R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the k-dimensional Hausdorff measure on the k-dimensional face on which it is concentrated. The remarkable fact is that a priori the directions of the faces are just Borel and no Lipschitz regularity is known. Notwithstanding that, we also prove that a Green-Gauss formula for these directions holds on special sets. 相似文献
5.
6.
Yaozhong Hu 《Journal of Theoretical Probability》1997,10(4):835-848
We give a formula of expanding the solution of a stochastic differential equation (abbreviated as SDE) into a finite Itô-Wiener chaos with explicit residual. And then we apply this formula to obtain several inequalities for diffusions such as FKG type inequality, variance inequality and a correlation inequality for Gaussian measure. A simple proof for Houdré-Kagan's variance inequality for Gaussian measure is also given. 相似文献
7.
Eberhard Siebert 《Journal of Theoretical Probability》1992,5(2):333-347
An operator-decomposable Gaussian measure on a separable Banach space can be factorized into a convolution product of a strongly operator-decomposable Gaussian measure and an operator-invariant Gaussian measure (with respect to the same operator). An example for this very factorization is discussed in some detail. In particular it is shown that a strongly operator-decomposable Gaussian measure need not necessarily be supported by the contraction subspace of the operator involved. Finally, the decomposability semigroup of a Gaussian measure turns out to be convex; and the corresponding invariance semigroup belongs to its extreme boundary. 相似文献
8.
In this paper we give a solution for the Gaussian version of the Busemann–Petty problem with additional information about dilates and translations. We also discuss the size of the Gaussian measure of the hyperplane sections of the dilates of the unit cube. 相似文献
9.
Mario Wschebor 《Stochastic Processes and their Applications》1983,14(2):147-155
A formula is proved for the expectation of the (d?1)-dimensional measure of the intersection of a Gaussian stationary random field with a fixed level u. 相似文献
10.
Hiroshi Sato 《Journal of Functional Analysis》1985,61(2):222-245
Let μ and μ1 be probability measures on a locally convex Hausdorff real topological linear space E. C. R. Baker (Lecture Notes in Mathematics No. 109, pp. 33–44, Springer-Verlag, Berlin/New York, 1979) posed the problem of characterizing the absolute continuity of μ and μ1 by their characteristic functionals. The aim of this paper is to give an answer to this problem in the case where μ is a Gaussian Radon measure. A Fourier transform shall be defined, the inversion formula established, and then a necessary and sufficient condition given for μ1 to be absolutely continuous with respect to μ based on the characteristic functional. As applications, for the convolution , where v is a Radon measure on E, we shall give some concrete sufficient conditions on v for . 相似文献
11.
From Crofton's formula for Minkowski tensors we derive stereological estimators of translation invariant surface tensors of convex bodies in the n‐dimensional Euclidean space. The estimators are based on one‐dimensional linear sections. In a design based setting we suggest three types of estimators. These are based on isotropic uniform random lines, vertical sections, and non‐isotropic random lines, respectively. Further, we derive estimators of the specific surface tensors associated with a stationary process of convex particles in the model based setting. 相似文献
12.
Let Ω ⊂ ℝd be a compact convex set of positive measure. A cubature formula will be called positive definite (or a pd-formula, for short)
if it approximates the integral ∫Ω f(x) dx of every convex function f from below. The pd-formulae yield a simple sharp error bound for twice continuously differentiable
functions. In the univariate case (d = 1), they are the quadrature formulae with a positive semidefinite Peano kernel of order
two. As one of the main results, we show that there is a correspondence between pd-formulae and partitions of unity on Ω.
This is a key for an investigation of pd-formulae without employing the complicated multivariate analogue of Peano kernels.
After introducing a preorder, we establish criteria for maximal pd-formulae. We also find a lower bound for the error constant
of an optimal pd-formula. Finally, we describe a phenomenon which resembles a property of Gaussian formulae. 相似文献
13.
This paper is concerned with various geometric averages of sections or projections of convex bodies. In particular, we consider Minkowski and Blaschke sums of sections as well as Minkowski sums of projections. The main result is a Crofton-type formula for Blaschke sums of sections. This is used to establish connections between the different averages mentioned above. As a consequence, we obtain results which show that, in some circumstances, a convex body is determined by the averages of its sections or projections.The research of the first author was supported in part by NSF grants DMS-9504249 and INT-9123373 相似文献
14.
In this paper we deal with strong Fenchel duality for infinite-dimensional optimization problems where both feasible set and objective function are evenly convex. To this aim, via perturbation approach, a conjugation scheme for evenly convex functions, based on generalized convex conjugation, is used. The key is to extend some well-known results from convex analysis, involving the sum of the epigraphs of two conjugate functions, the infimal convolution and the sum formula of ??-subdifferentials for lower semicontinuous convex functions, to this more general framework. 相似文献
15.
Alexander Koldobsky 《Advances in Mathematics》2003,177(1):105-114
The Busemann-Petty problem asks whether symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller n-dimensional volume. The solution has recently been completed, and the answer is affirmative if n?4 and negative if n?5. In this article we present a short proof of the affirmative result and its generalization using the Funk-Hecke formula for spherical harmonics. 相似文献
16.
设Y是局部凸向量空间,其上装配有GaussianRadon测度γ.A(Y)(或ε(Y)是Y上检验函数空间(或με(Y)是相应的分布函数空间·我们证明了:(或με(Y),并由此得到μA(Y)(或με(Y))上的Fourier变换公式.其中“*”表示复共轭算子,“”表示连续稠线性嵌入.进一步还得到了A(Y)(或ε(Y))上无穷维伪微分算子A是L2(Y,γ)上连续的充要条件是其共轭算子A’满足A’(L2(Y,γ)L2(Y,γ). 相似文献
17.
The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including
a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections ofl
p-balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts
of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of
the recent result of Barthe and Naor on extremal projections ofl
p-balls, and give a Fourier analytic solution to Shephard’s problem, originally solved by Petty and Schneider and asking whether
symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula
expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function. 相似文献
18.
Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce two relatives of the covering parameter called covering index and weak covering index, which upper bound well-studied quantities like the illumination number, the illumination parameter and the covering parameter of a convex body. Intuitively, the two indices measure how well a convex body can be covered by a relatively small number of homothets having the same relatively small homothety ratio. We show that the covering index is a lower semicontinuous functional on the Banach-Mazur space of convex bodies. We further show that the affine d-cubes minimize the covering index in any dimension d, while circular disks maximize it in the plane. Furthermore, the covering index satisfies a nice compatibility with the operations of direct vector sum and vector sum. In fact, we obtain an exact formula for the covering index of a direct vector sum of convex bodies that works in infinitely many instances. This together with a minimization property can be used to determine the covering index of infinitely many convex bodies. As the name suggests, the weak covering index loses some of the important properties of the covering index. Finally, we obtain upper bounds on the covering and weak covering index. 相似文献
19.
Yu. Ch. Kokaev 《Journal of Mathematical Sciences》1984,27(5):3084-3094
One investigates the question of the exact bounds of the distribution of an arbitrary Gaussian linear measurable functional with respect to some conditional Gaussian measure, generated by a convex subset of a space with a Gaussian measure.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 128–143, 1982, 相似文献
20.
We provide a sharp quantitative version of the Gaussian concentration inequality: for every \(r>0\), the difference between the measure of the r-enlargement of a given set and the r-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also prove a similar estimate in the Euclidean setting for the enlargement with a general convex set. This is equivalent to the stability of the Brunn–Minkowski inequality for the Minkowski sum between a convex set and a generic one. 相似文献