On Separation of Points from Additive Subgroups of Banach Spaces by Continuous Characters and Positive Definite Functions

Let G be an additive subgroup of a normed space X. We say that a point is weakly separated (resp. -separated) from G if it can be separated from G by a continuous character (resp. by a continuous positive definite function). Let T : X → Y be a continuous linear operator. Consider the following conditions: (ws) if , then x is weakly separated from G; (ps) if , then x is -separated from G; (wp) if Tx is -separated from T(G), then x is weakly separated from G. By (resp. , ) we denote the class of operators T : X → Y which satisfy (ws) (resp. (ps), (wp)) for all and all subgroups G of X. The paper is an attempt to describe the above classes of operators for various Banach spaces X, Y. It is proved that if X, Y are Hilbert spaces, then is the class of Hilbert-Schmidt operators. It is also shown that if T is a Hilbert-to-Banach space operator with finite ℓ-norm, then .      

The Dvoretzky-Hanani Theorem for the Group "ax + b"

Let G be the group "ax + b" of affine transformations of the line and let U be a neighbourhood of 1 in G. It is proved that there is another neighborhood V of 1 such that to each finite sequence g1,...,gn V there corresponds a sequence of signs 1,...,n = ±1 with U for k = 1,...,n. This implies that G satisfies the following analogue of the Dvoretzky-Hanani theorem: to each sequence converging to 1 in G there corresponds a sequence of signs k = ±1 such that the infinite product is convergent.     

Balancing vectors and Gaussian measures of n-dimensional convex bodies

Let ‖·‖ be the Euclidean norm on R ⁿ and γ_n the (standard) Gaussian measure on R ⁿ with density (2π)^−n/2e. It is proved that there is a numerical constant c>0 with the following property: if K is an arbitrary convex body in R ⁿ with γ_n(K)≥1/2, then to each sequence u₁,…,u_m∈ R ⁿ with ‖u₁‖,…,‖u_m‖≤c there correspond signs ε₁,…,ε_m=±1 such that ∑^m_i=1ε_iu_i∈K. This improves the well-known result obtained by Spencer [Trans. Amer. Math. Soc. 289 , 679–705 (1985)] for the n-dimensional cube. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 351–360, 1998     

On series of signed vectors and their rearrangements

Let x₁,…,x_m∈ \input amssym $ \Bbb R$ n be a sequence of vectors with ∥x_i∥₂ ≤ 1 for all i. It is proved that there are signs ε₁,…,ε_m = ±1 such that where C₁, C₂ are some numerical constants. It is also proved that there are signs ε,…,ε = ±1 and a permutation π of {1,…,m} such that where C^′ is some other numerical constant. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Inequalities for convex bodies and polar reciprocal lattices inR n II: Application ofK-convexity

The paper is a supplement to [2]. LetL be a lattice andU ano-symmetric convex body inR ⁿ . The Minkowski functional? ⁿ ofU, the polar bodyU ⁰, the dual latticeL ^*, the covering radius μ(L, U), and the successive minima λ _i ,i=1, …,n, are defined in the usual way. Let $\mathcal{L}_n $ be the family of all lattices inR ⁿ . Given a convex bodyU, we define $$\begin{gathered} mh(U){\text{ }} = {\text{ }}\sup {\text{ }}\max \lambda _i (L,U)\lambda _{n - i + 1} (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n 1 \leqslant i \leqslant n \hfill \\ lh(U){\text{ }} = {\text{ }}\sup {\text{ }}\lambda _1 (L,U) \cdot \mu (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n \hfill \\ \end{gathered} $$ and kh(U) is defined as the smallest positive numbers for which, given arbitrary $L \in \mathcal{L}_n $ andx∈R ⁿ /(L+U), somey∈L ^* with ∥y∥ _U ⁰?sd(xy,Z) can be found. It is proved $$C_1 n \leqslant jh(U) \leqslant C_2 nK(R_U^n ) \leqslant C_3 n(1 + \log n),$$ , for j=k, l, m, whereC ₁,C ₂,C ₃ are some numerical constants andK(R _U ⁿ ) is theK-convexity constant of the normed space (R ⁿ , ∥∥U). This is an essential strengthening of the bounds obtained in [2]. The bounds for lh(U) are then applied to improve the results of Kannan and Lovász [5] estimating the lattice width of a convex bodyU by the number of lattice points inU.