The paper is a supplement to [2]. Let L be a lattice and U an o-symmetric convex body inR n. The Minkowski functional I[ I1~ of U, the polar body U ~ the dual lattice L*, the covering radius/z(L, U), and the successive minima ~.i(L, U), i = 1 . . . . . n, are defined in the usual way. Let 12, be the family of all lattices in R". Given a convex body U, we define mh(U) = sup max ~.i(L, U)~.n-i+l(...

A function f is said to be cone superadditive if there exists a partition of R into a family of polyhedral convex cones such that f(z + x) + f(z + y) ≤ f(z) + f(z + x+ y) holds whenever x and y belong to the same cone in the family. This concept is useful in nonlinear integer programming in that, if the objective function is cone superadditive, the global minimality can be characterized by loca...

1. Introduction and notation. In this paper the word " local " is used in at least three different meanings. Our aim is to study local Banach spaces of Fréchet or other locally convex spaces, and it turns out that it is convenient to use the local theory of Banach spaces for this purpose. Recall that given a locally convex space E and a continuous seminorm p on E the completion of the normed sp...

New results on approximation of a convex body K ⊂ R3 by affine images of circular cylinders, parallelepipeds, hexagonal and octagonal regular (and some other) prisms are obtained. Two of the theorems obtained are as follows (V (K) denotes the volume of a body K ⊂ R3). Theorem 1. Let K be an arbitrary convex body in R3. There exists a regular octagonal prism an affine image of which is circumscr...

This paper is to characterize L-concavities and L-convexities via some derived forms of relations operators. Specifically, notions L-concave internal relation space hull are introduced. It proved that the category L-concave
 spaces isomorphic spaces. Also, L-convex enclosed

Boundary effects are crucial for dynamics of dilute charged gases governed by the Vlasov-Poisson-Boltzmann (VPB) system. In this paper, we study existence and regularity solutions to VPB system with soft potential in a bounded convex domain in-flow boundary condition. We establish strong time interval $[0,T]$ an arbitrary given $T>0$ when initial distribution function is near absolute Maxwellia...

in this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. furthermore,...

For functions defined on integer lattice points, discrete versions of the Hessian matrix have been considered in various contexts. In discrete convex analysis, for example, certain combinatorial properties of the discrete Hessian matrices are known to characterize M-convex and L-convex functions, which can be extended to convex functions in real variables. The relationship between convex extens...

The state estimation of a quantized system (Q.S.) is a challenging problem for designing feedback control and model-based fault diagnosis algorithms. The core of a Q.S. is a continuous variable system whose inputs and outputs are represented by their corresponding quantized values. This paper concerns with state estimation of a Q.S. by a qualitative observer. The presented observer in this pape...