Centrally symmetric convex bodies and sections having maximal quermassintegrals

Let d ≧ 2, and let K ⊂ ℝd be a convex body containing the origin 0 in its interior. In a previous paper we have proved the following. The body K is 0-symmetric if and only if the following holds. For each ω ∈ Sd−1, we have that the (d − 1)-volume of the intersection of K and an arbitrary hyperplane, with normal ω, attains its maximum if the hyperplane contains 0. An analogous theorem, for 1-dimensional sections and 1-volumes, has been proved long ago by Hammer (see [2]). In this paper we deal with the ((d − 2)-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small C2-perturbations, or C3-perturbations of the Euclidean unit ball, respectively.

Multidimensional local theorem in the Knopfmachers semigroup

In the presentpaper a multidimensionallocal theorem for arithmetic functions definedin the Knopfmachers semigroup G is obtained.

The Local Theorem for Monotypic Tilings

A locally finite face-to-face tiling $\cal T$ of euclidean $d$-space ${\Bbb E}^d$ is monotypic if each tile of $\cal T$ is a convex polytope combinatorially equivalent to a given polytope, the combinatorial prototile of $\cal T$. The paper describes a local characterization of combinatorial tile-transitivity of monotypic tilings in ${\Bbb E}^d$; the result is the Local Theorem for Monotypic Tilings. The characterization is expressed in terms of combinatorial symmetry properties of large enough neighborhood complexes of tiles. The theorem sits between the Local Theorem for Tilings, which describes a local characterization of isohedrality (tile-transitivity) of monohedral tilings (with a single isometric prototile) in ${\Bbb E}^d$, and the Extension Theorem, which gives a criterion for a finite monohedral complex of polytopes to be extendable to a global isohedral tiling of space.

Moving Ergodic Theorems for Superadditive Processes

Letbe a semigroup of measure preserving transformations on a measure space (Ω, ℱ, μ). The main result of the paper is the proof of a.e. convergence for the moving averageswhere {FIn} is a superadditive process and {In} is a sequence of cubes insatisfying the "cone-condition". The identification of the limit is given. A moving local theorem is also proved.

Semi-Groups in L∞ And Local Ergodic Theorem

We show that any W*-continuous semi-group in L∞ is L1-norm continuous. As an application we prove the n-dimensional local ergodic theorem in L∞. We also note that any bounded additive process in L∞ is absolutely continuous.For n = 1 this local theorem improves those of R. Sato [14] and D. Feyel [6] and for n ≥ 1 it generalizes M. Lin's ones which hold for positive operators [12].