Support and separation properties of convex sets in finite dimension

Separation of Convex Sets

A line L separates a set A from a collection S of plane sets if A is contained in one of the closed half-planes defined by L, while every set in S is contained in the complementary closed half-plane. Let f(n) be the largest integer such that for any collection F of n closed disks in the plane with pairwise disjoint interiors, there is a line that separates a disk in F from a subcollection of F ...

Characterizations of Convex Sets by Local Support Properties

It is our purpose to establish some new characterizations of convex sets by means of local properties and to derive as a consequence certain known results. This will be done for sets in a topological linear space 7, such a space being a real linear space with a Hausdorff topology such that the operations of vector addition x+y and scalar multiplication ax are continuous in both variables jointl...

Finite Sets as Complements of Finite Unions of Convex Sets

Suppose S ⊆ R is a set of (finite) cardinality n whose complement can be written as the union of k convex sets. It is perhaps intuitively appealing that when n is large k must also be large. This is true, as is shown here. First the case in which the convex sets must also be open is considered, and in this case a family of examples yields an upper bound, while a simple application of a theorem ...

dedekind modules and dimension of modules

در این پایان نامه، در ابتدا برای مدول ها روی دامنه های پروفر شرایط معادل به دست آورده ایم و خواصی از ددکیند مدول ها روی دامنه های پروفر مشخص کرده ایم. در ادامه برای ددکیند مدول های با تولید متناهی روی حلقه های به طور صحیح بسته شرایط معادل به دست آورده ایم و ددکیند مدول های ضربی را مشخص کرده ایم. گزاره هایی در مورد بعد ددکیند مدول ها بیان کرده ایم. در پایان، قضایای lying over و going down را برا...

Separation Properties of Convex Cones1