We show that a PL-realization of a closed connected manifold of dimension n − 1 in R (n ≥ 3) is the boundary of a convex polyhedron if and only if the interior of each (n− 3)-face has a point, which has a neighborhood lying on the boundary of a convex n-dimensional body. This result is derived from a generalization of Van Heijenoort’s theorem on locally convex manifolds to the spherical case. O...

Let F be a family of sets in Rd (always d≥2). A set M⊂Rd is called F-convex, if for any pair distinct points x,y∈M, there F∈F, such that x,y∈F and F⊂M. four {w,x,y,z}⊂Rd rectangular quadruple, conv{w,x,y,z} non-degenerate rectangle. If the all quadruples, then we obtain right quadruple convexity, abbreviated as rq-convexity. In this paper focus on rq-convexity complements, taken most cases ball...

Inspired by locale theory, we propose “pointfree convex geometry.” We introduce the notion of convexity algebra as pointfree convexity space. There are two notions of point for convexity algebra: one is chain-prime meet-complete filter and the other is maximal meet-complete filter. In this paper we show the following: (1) the former notion of point induces a dual equivalence between the categor...

Many optimization problems in queueing theory have functions with integer and real variables. Convexity results for these functions can be obtained by fixing either the integer or the real variable where bounds of the considered function play an important role. Tokgöz, Maalouf and Kumin [11] introduces the concept of mixed convexity for functions with real and integer variables and obtain conve...

A new light is shed on “substitutes and complements” in the maximum weight circulation problem with reference to the concepts of L-convexity and M-convexity in the theory of discrete convex analysis. This provides us with a deeper understanding of the relationship between convexity and submodularity in combinatorial optimization.

This is a survey of the results in [1, 2, 3, 4, 5, 6] on convexity numbers of closed sets in Rn, homogeneity numbers of continuous colorings on Polish spaces and families of functions covering Rn. 1. Convexity numbers of closed sets in R A natural way to measure the degree of non-convexity of a subset S of a real vector space is the convexity number γ(S), the least size of a family C of convex ...

Congestion games have extensively been studied till recently. An interesting special class of congestion games on networks is the class of congestion games on extension-parallel networks considered by Holzman and Law-yone (2003). It is shown by Fotakis (2010) that for every congestion game on an extension-parallel network, any best-response sequence reaches a pure Nash equilibrium of the game i...