The Weber Problem for a given nite set of existing facilities Ex = fEx 1 ; Ex 2 ; : : : ; Ex M g IR 2 with positive weights w m (m = 1; : : : ; M) is to nd a new facility X such that P M m=1 w m d(X; Ex m) is minimized for some distance function d. A variation of this problem is obtained if the existing facilities are situated on two sides of a linear barrier. Such barriers like rivers, highway...

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 ${M_{n,m}}$.

In this work, we introduce the class of h-MN-convex functions by generalizing the concept of MN-convexity and combining it with h-convexity. Namely, let M : [0, 1] → [a, b] be a Mean function given by M (t) = M (t; a, b); where by M (t; a, b) we mean one of the following functions: At (a, b) := (1− t) a + tb, Gt (a, b) = a1−tbt and Ht (a, b) := ab ta+(1−t)b = 1 At( 1 a , 1 b ) ; with the proper...

An explicit expression is obtained for the perimeter and area generating function G(y, z) = ∑ n>=2 ∑ m>=1 cn,my z, where cn,m is the number of row-convex polygons with area m and perimeter n. A similar expression is obtained for the area-perimeter generating function for staircase polygons. Both expressions contain q-series.