Let K ⊂ R d be a convex body, and assume that L is randomly rotated shifted integer lattice. the hull of (random) points ∩ . The mean width W ( ) investigated. asymptotic order difference λ − maximized by obtained polytopes minimized for smooth sets as → ∞

Two lattice points are visible to one another if there exist no other on the line segment connecting them. In this paper we study convex polygons that contain a point such all in polygon from it. We completely classify polygons, show finitely many of width greater than 2, and computationally enumerate As an application classification, prove new obstructions graphs arising as skeleta tropical pl...

We continue 3], the study of the lattice L n generated by cuts of the complete graph on a set V n of n vertices. The lattice L n spans an N = ? n 2-dimensional space of all functions deened on a set V 2 n of all unordered pairs of the set V n. We prove that the cut polytope, i.e. the convex hull of all cuts, is an asymmetric Delaunay polytope of L n. Symmetric Delaunay polytopes of a lattice L ...

a subgroup $x$ of a group $g$ is almost normal if the index $|g:n_g(x)|$ is finite, while $x$ is nearly normal if it has finite index in the normal closure $x^g$. this paper investigates the structure of groups in which every (infinite) subgroup is either almost normal or nearly normal.

Introduction. We propose to investigate here the consequences of the identity of each pair chosen from three important generalizations of the relation of betweenness on a line, namely, algebraic betweenness [l, p. 27 J, metric betweenness [3, p. 36], and lattice betweenness [7, Part I I ] . We shall also find an interpretation of metric betweenness in the Banach space of all continuous function...

The existence of lateral completions of `-groups is an old problem that was first solved, for conditionally complete vector lattices, by Nakano [5]. The existence and uniqueness of lateral completions of representable `-groups was first obtained as a consequence of the orthocompletions of Bernau [1], and later the proofs were simplified by Conrad [3], who also proved the existence and uniquenes...

A locally modular (resp. locally distributive) lattice is a lattice with a congruence relation and each of whose equivalence class has sufficiently many elements and is a modular (resp. distributive) sublattice. Both the lattice of all closed subspaces of a locally convex space and the lattice of projections of a locally finite von Neumann algebra are locally modular. The lattice of all /^-topo...

We have already considered instances of the following type of problem: given a bounded subset Ω of Euclidean space R N , to determine #(Ω ∩ Z N), the number of integral points in Ω. It is clear however that there is no answer to the problem in this level of generality: an arbitrary Ω can have any number of lattice points whatsoever, including none at all. In [Gauss's Circle Problem], we counted...

The theory of fuzzy sets was inspired by Zadeh [10]. Subsequently, Rosenfeld introduced the concept of a fuzzy subgroup of a group [9]. Fuzzy cosets and fuzzy normal subgroups of a group G have been studied in [3, 5, 8]. In [4], the ring of cosets of a fuzzy ideal was constructed. Let A and B be fuzzy subgroups of G such that B ⊆ A. The purpose of this paper is to introduce the notion of fuzzy ...