We introduce a new operation for the difference of two sets A and C of Rn depending on a parameter . This new operation may yield as special cases the classical difference and the Minkowski difference, if the sets A and C are closed, convex sets, if int(C) is nonempty, and if A or C bounded. Continuity properties with respect to both the operands and the parameter of this operation are studied. Lipschitz properties of the Minkowski difference between two sets of a normed vector space are proved in the bounded case as well as in the unbounded case without condition on the dimension of the space. 相似文献
Let X be a real Banach space. Let be a family of closed, convex subsets of X. We show that either the intersection ?γ∈Γ(Gγ)? of the ?-neighborhood of the sets Gγ is bounded for each ?>0, or it is unbounded for each ?>0. From this we derive a fixed point theorem for suitable maps that move some points along a bounded direction in Hilbert spaces. 相似文献
We show that with the weak topology is not an intersection of Borel sets in its Cech-Stone extension (and hence in any compactification). Assuming (CH), this implies that has no continuous injection onto a Borel set in a compact space, or onto a Lindelöf space. Under (CH), this answers a question of Arhangel'ski.
We study category counterparts of the notion of a universal measure zero set of reals.
We say that a set is universally meager if every Borel isomorphic image of is meager in . We give various equivalent definitions emphasizing analogies with the universally null sets of reals.
In particular, two problems emerging from an earlier work of Grzegorek are solved.
We define a new notion of ``HP-small' set which implies that is both -porous and Haar null in the sense of Christensen. We show that the set of all continuous functions on which have finite unilateral approximate derivative at a point is HP-small, as well as its projections onto hyperplanes. As a corollary, the same is true for the set of all Besicovitch functions. Also, the set of continuous functions on which are Hölder at a point is HP-small.
A normed and partially ordered vector space of so-called directed sets is constructed, in which the convex cone of all nonempty convex compact sets in Rn is embedded by a positively linear, order preserving and isometric embedding (with respect to a new metric stronger than the Hausdorff metric and equivalent to the Demyanov one). This space is a Banach and a Riesz space for all dimensions and a Banach lattice for n=1. The directed sets in Rn are parametrized by normal directions and defined recursively with respect to the dimension n by the help of a support function and directed supporting faces of lower dimension prescribing the boundary. The operations (addition, subtraction, scalar multiplication) are defined by acting separately on the support function and recursively on the directed supporting faces. Generalized intervals introduced by Kaucher form the basis of this recursive approach. Visualizations of directed sets will be presented in the second part of the paper. 相似文献
We study the asymptotic behavior and limit distributions for sums Sn =bn-1i=1ni,where
i, i 1, are i.i.d. random convex compact (cc) sets in a given separable Banach space B and summation is defined in a sense of Minkowski. The following results are obtained: (i) Series (LePage type) and Poisson integral representations of random stable cc sets in B are established; (ii) The invariance principle for processes Sn(t) =bn-1i=1[nt]i, t[0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where
i are segments, the limit of Sn is proved to be countable zonotope. Furthermore, if B = Rd, the singularity of distributions of two countable zonotopes Yp1, 1,Yp2, 2, corresponding to values of exponents p1, p2 and spectral measures
1,
2, is proved if either p1p2 or
12; (iv) Some new simple estimates of parameters of stable laws in Rd, based on these results are suggested. 相似文献
In this article we study minimal1-blocking sets in finite projective spaces PG(n,q),n 3. We prove that in PG(n,q2),q = ph, p prime, p > 3,h 1, the second smallest minimal 1-blockingsets are the second smallest minimal blocking sets, w.r.t.lines, in a plane of PG(n,q2). We also study minimal1-blocking sets in PG(n,q3), n 3, q = ph, p prime, p > 3,q 5, and prove that the minimal 1-blockingsets of cardinality at most q3 + q2 + q + 1 are eithera minimal blocking set in a plane or a subgeometry PG(3,q). 相似文献
We present the Komlós theorem for multivalued functions whose values are closed (possibly unbounded) convex subsets of a separable Banach space. Komlós theorem can be seen as a generalization of the SLLN for it deals with a sequence of integrable multivalued functions that do not have to be identically distributed nor independent. The Artstein–Hart SLLN for random sets with values in Euclidean spaces is derived from the main result. Finally, since the main theorem concerns multifunctions whose values are allowed to be unbounded, we can restate it in terms of normal integrands (random lower semicontinuous functions). 相似文献