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1.
In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body K ? Rd has the property that the tangent cone of every non-smooth point q ? ?K is acute (in a certain sense), then there is a closed billiard trajectory in K.  相似文献   

2.
Let #K be a number of integer lattice points contained in a set K. In this paper we prove that for each d ∈ N there exists a constant C(d) depending on d only, such that for any origin-symmetric convex body K ? R d containing d linearly independent lattice points
$$\# K \leqslant C\left( d \right)\max \left( {\# \left( {K \cap H} \right)} \right)vo{l_d}{\left( K \right)^{\frac{{d - m}}{d}}},$$
where the maximum is taken over all m-dimensional subspaces of R d . We also prove that C(d) can be chosen asymptotically of order O(1) d d d?m . In particular, we have order O(1) d for hyperplane slices. Additionally, we show that if K is an unconditional convex body then C(d) can be chosen asymptotically of order O(d) d?m .
  相似文献   

3.
In this note, we give a necessary and sufficient condition for viability property of diffusion processes with jumps on closed submanifolds of R m . Our result is the system is viable in a closed submanifold K iff the coefficients are tangent to K along K if the equation is in the sense of stratonovich integral and the solution jumps from K to K.  相似文献   

4.
In the paper, a formula to calculate the probability that a random segment L(ω, u) in R n with a fixed direction u and length l lies entirely in the bounded convex body D ? R n (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ? D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ? D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane.  相似文献   

5.
Many results in Combinatorial Integral Geometry are derived by integration of the combinatorial decompositions associated with finite point sets {P i } given in the plane ?2. However, most previous cases of integration of the decompositions in question were carried out for the point sets {P i } containing no triads of collinear points, where the familiar algorithm sometimes called the “Four indicator formula” can be used. The present paper is to demonstrate that the complete combinatorial algorithm valid for sets {P i } not subject to the mentioned restriction opens the path to various results, including the field of Stochastic Geometry. In the paper the complete algorithm is applied first in an integration procedure in a study of the perforated convex domains, i.e convex domains containing a finite array of non-overlapping convex holes. The second application is in the study of random colorings of the plane that are Euclidean motions invariant in distribution, basing on the theory of random polygonal windows from the so-called Independent Angles (IA) class. The method is a direct averaging of the complete combinatorial decompositions written for colorings observed in polygonal windows from the IA class. The approach seems to be quite general, but promises to be especially effective for the random coloring generated by random Poisson polygon process governed by the Haar measure on the group of Euclidean motions of the plane, assuming that a point P ∈ ?2 is colored J if P is covered by exactly J polygons of the Poisson process. A general theorem clearing the way for Laplace transform treatment of the random colorings induced on line segments is formulated.  相似文献   

6.
Let K and L be two convex bodies in R4, such that their projections onto all 3-dimensional subspaces are directly congruent. We prove that if the set of diameters of the bodies satisfies an additional condition and some projections do not have certain π-symmetries, then K and L coincide up to translation and an orthogonal transformation. We also show that an analogous statement holds for sections of star bodies, and prove the n-dimensional versions of these results.  相似文献   

7.
The author has established that if [λn] is a convex sequence such that the series Σn -1λn is convergent and the sequence {K n} satisfies the condition |K n|=O[log(n+1)]k(C, 1),k?0, whereK n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a n}, then the series Σlog(n+1)1-kλn a n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].  相似文献   

8.
We present necessary and sufficient conditions on planar compacta K and continuous functions f on K in order that f generate the algebras P(K), R(K), A(K) or C(K). We also unveil quite surprisingly simple examples of non-polynomial convex compacta K ? C and fP(K) with the property that fP(K) is a homeomorphism of K onto its image, but for which f ?1 ? P(f(K)). As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull \(\widehat K\). On the other hand, it is shown that the restriction f*|G of the Gelfand-transform f* of an injective function fP(K) is injective on every regular, bounded complementary component G of K. A necessary and sufficient condition in terms of the behaviour of f on the outer boundary of K is given in order that f admit a holomorphic injective extension to \(\widehat K\). We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.  相似文献   

9.
Any (measurable) function K from Rn to R defines an operator K acting on random variables X by K(X) = K(X1,..., Xn), where the Xj are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x1, x2,..., xn) ∈ {x1, x2,..., xn}. For each such continuous selector H (except for projections onto a single coordinate) there is a unique point ωH in the interval (0, 1) so that, for any random variable X, the iterates H(N) acting on X converge in distribution as N → ∞ to the ωH-quantile of X.  相似文献   

10.
An isoperimetric inequality and a Poincare-type inequality are proved for probability measures on the line that are the images of a uniform distribution on a convex compact subset of R n under polynomial mappings of fixed degree d.  相似文献   

11.
Necessary conditions are established for the continuity of finite sums of ridge functions defined on convex subsets E of the space Rn. It is shown that under some constraints imposed on the summed functions ?i, in the case when E is open, the continuity of the sum implies the continuity of all ?i. In the case when E is a convex body with nonsmooth boundary, a logarithmic estimate is obtained for the growth of the functions ?i in the neighborhoods of the boundary points of their domains of definition. In addition, an example is constructed that demonstrates the accuracy of the estimate obtained.  相似文献   

12.
Combinatorial integral geometry possesses some results that can be interpreted as belonging to the field of Geometric Tomography. The main purpose of the present paper is to present a case of parallel X-ray approach to tomography of random convex polygons. However, the Introduction reviews briefly some earlier results by the author that refer to reconstruction of (non-random) convex domains by means of a point X-ray. The main tool in treating the parallel X-rays is disintegrated Pleijel identity, or rather, its averaged version, whose derivation is represented in complete detail. The paper singles out a class of random polygons called tomography models, that offer essential advantages for the analysis. The definition of a tomography model is given in terms of stochastic independence. Fortunately, random translation-invariant Poisson processes of lines in IR 2 suggest a class of examples. We recall that each such line process is determined by its rose of directions ρ(?). For rather general ρ(?), the number weighted typical polygon in the polygonal partition of the plane generated by the corresponding Poisson line process happens to be a tomography model. For general tomography models, a differential equation is derived for the Laplace transform for parallel X-rays, that rises several interesting computational problems.  相似文献   

13.
We say that a convex set K in ? d strictly separates the set A from the set B if A ? int(K) and B ? cl K = ø. The well-known Theorem of Kirchberger states the following. If A and B are finite sets in ? d with the property that for every T ? A?B of cardinality at most d + 2, there is a half space strictly separating T ? A and T ? B, then there is a half space strictly separating A and B. In short, we say that the strict separation number of the family of half spaces in ? d is d + 2.In this note we investigate the problem of strict separation of two finite sets by the family of positive homothetic (resp., similar) copies of a closed, convex set. We prove Kirchberger-type theorems for the family of positive homothets of planar convex sets and for the family of homothets of certain polyhedral sets. Moreover, we provide examples that show that, for certain convex sets, the family of positive homothets (resp., the family of similar copies) has a large strict separation number, in some cases, infinity. Finally, we examine how our results translate to the setting of non-strict separation.  相似文献   

14.
We study the number of k-element sets A? {1,...,N} with |A+A| ≤ K|A| for some (fixed) K > 0. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a factor of 2 o ( k ) N o (1) for most N and k. As a consequence of this and a further new result concerning the number of sets A??/N? with |A+A| ≤ c|A|2, we deduce that the random Cayley graph on ?/N? with edge density ½ has no clique or independent set of size greater than (2+o(1)) log2 N, asymptotically the same as for the Erd?s-Rényi random graph. This improves a result of the first author from 2003 in which a bound of 160log2 N was obtained. As a second application, we show that if the elements of A ? ? are chosen at random, each with probability 1/2, then the probability that A+A misses exactly k elements of ? is equal to (2+O(1))?k/2 as k → ∞.  相似文献   

15.
The minimum size of a binary code with length n and covering radius R is denoted by K(n, R). For arbitrary R, the value of K(n, R) is known when n ≤  2R +  3, and the corresponding optimal codes have been classified up to equivalence. By combining combinatorial and computational methods, several results for the first open case, K(2R +  4, R), are here obtained, including a proof that K(10, 3) =  12 with 11481 inequivalent optimal codes and a proof that if K(2R +  4, R) <  12 for some R then this inequality cannot be established by the existence of a corresponding self-complementary code.  相似文献   

16.
In this paper we prove the following conformity criterion for the gradient of conformal radius ?R(D, z) of a convex domain D: the boundary ?D has to be a circumference. We calculate coefficients K(r) for K(r)-quasiconformal mappings ?R(D(r), z), D(r) ? D, 0 < r < 1, and complete the results obtained by F. G. Avkhadiev and K.-J. Wirths for the structure of boundary elements of quasiconformal mappings of the domain D.  相似文献   

17.
The symmetry of convex bodies of constant width is discussed in this paper. We proved that for any convex body K?? n of constant width, \(1\leq \mathrm{as}_{\infty}(K)\leq\frac{n+\sqrt{2n(n+1)}}{n+2}\), where as(?) denotes the Minkowski measure of asymmetry for convex bodies. Moreover, the equality holds on the left-hand side precisely iff K is an Euclidean ball and the upper bounds are attainable, in particular, if n=3, the equality holds on the right-hand side if K is a Meissner body.  相似文献   

18.
We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on Z. Select vertical columns at random independently with a given positive probability. Keep (respectively remove) vertical edges in the selected columns, with probability p (respectively 1?p). All horizontal edges and vertical edges lying in unselected columns are kept (respectively removed) with probability q (respectively 1 ? q). We show that, if p > pc(Z2) (the critical point for homogeneous Bernoulli bond percolation), then q can be taken strictly smaller than pc(Z2) in such a way that the probability that the origin percolates is still positive.  相似文献   

19.
The main result of this paper asserts that the distribution density of any non-constant polynomial f12,...) of degree d in independent standard Gaussian random variables ξ1 (possibly, in infinitely many variables) always belongs to the Nikol’skii–Besov space B1/d (R1) of fractional order 1/d (see the definition below) depending only on the degree of the polynomial. A natural analog of this assertion is obtained for the density of the joint distribution of k polynomials of degree d, also with a fractional order that is independent of the number of variables, but depends only on the degree d and the number of polynomials. We also give a new simple sufficient condition for a measure on Rk to possess a density in the Nikol’skii–Besov class Bα(R)k. This result is applied for obtaining an upper bound on the total variation distance between two probability measures on Rk via the Kantorovich distance between them and a certain Nikol’skii–Besov norm of their difference. Applications are given to estimates of distributions of polynomials in Gaussian random variables.  相似文献   

20.
The present paper studies the following constrained vector optimization problem: \(\mathop {\min }\limits_C f(x),g(x) \in - K,h(x) = 0\), where f: ? n → ? m , g: ? n → ? p are locally Lipschitz functions, h: ? n → ? q is C 1 function, and C ? ? m and K ? ? p are closed convex cones. Two types of solutions are important for the consideration, namely w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point x 0 to be a w-minimizer and first-order sufficient conditions for x 0 to be an i-minimizer are obtained. Their effectiveness is illustrated on an example. A comparison with some known results is done.  相似文献   

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