The problem of minimizing f̃ = f +p over some convex subset of a Euclidean space is investigated, where f(x) = x Ax + b x is a strictly convex quadratic function and |p| is only assumed to be bounded by some positive number s. It is shown that the function f̃ is strictly outer γ-convex for any γ > γ∗, where γ∗ is determined by s and the smallest eigenvalue of A. As consequence, a γ∗-local minimal...

Let X rv Np(O, E), the p-variate normal distribution with mean °and positive definite covariance matrix .E. Anderson (1955) showed that if .E2 .E1 is positive semidefinite then PEl (C) 2: P E2(C) for every centrally symmetric (-C = convex set C ~ RP. Fefferman, Jodeit, and Perlman (1972) extended this result to elliptically contoured distributions. In the present paper similar multivariate f f"...

Let E(k, l) denote the smallest integer such that any set of at least E(k, l) points in the plane, no three on a line, contains either an empty convex polygon with k vertices or an empty pseudo-triangle with l vertices. The existence of E(k, l) for positive integers k, l ≥ 3, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565–576, 2007]. In this pa...

The problem of optimizing a biconvex function over a given (bi)convex or compact set frequently occurs in theory as well as in industrial applications, for example, in the field of multifacility location or medical image registration. Thereby, a function f : X×Y → R is called biconvex, if f(x, y) is convex in y for fixed x ∈ X, and f(x, y) is convex in x for fixed y ∈ Y . This paper presents a ...

Let X be a nonempty, compact, convex set in R and let f be an upper semicontinuous mapping from X to the collection of nonempty, compact, convex subsets of R. It is well known that such a mapping has a stationary point on X; i.e., there exists a point X such that its image under f has a nonempty intersection with the normal cone of X at the point. In the case where, for every point in X, it hol...

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...

In this paper we study the concept of Latin-majorizati-\on. Geometrically this concept is different from other kinds of majorization in some aspects. Since the set of all $x$s Latin-majorized by a fixed $y$ is not convex, but, consists of union of finitely many convex sets. Next, we hint to linear preservers of Latin-majorization on $ mathbb{R}^{n}$ and ${M_{n,m}}$.

Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs Gn : F = (Gn)n ≥ 1 depending on n as follows: the order |V (G)| = φ(n) and lim n→∞ φ(n) = ∞. If there exists a constant C > 0 such that dim(Gn) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension, otherwise F has unbou...