Konveks geometride karakterizasyonlar

Abstract

In this thesis starshaped sets which are one of the variants of convex sets are examined. A property referred to points of a certain set S is a local property if it relates any point with points included in a certain neigborhood of x. Otherwise, we speak of a global property. For example, convexity or starshapedness are global properties whereas the condition of being a boundary point is a local property. The basic aim of this thesis is to provide some formulations for representations of convex kernel which is a basic problem of characterizations of starshapedness of a set. In the first chapter basic concepts of convex sets are given. In the second chapter some basic characterizations of convex sets are given and some weakened convexity types are listed. In the third chapter some structural properties of starshaped sets and some properties of visibility geometry are examined. In the fourth chapter starshapedness theory is examined which is based on a survey article of Fausto A. Toranzos and in the last chapter our results are given; The first two result depend on the concept of maximal convex components. First result is a characterization of the convex kernel of S by means of halfspaces to maximal convex subsets in closure of complement of S ( ). We give also a formula for the convex kernel of some as intersections of the sets in the S such that every maximal convex subset of deficiency of S admit its boundary relative extreme subset with respect to S. Some other formulas for the kernel are given by using points of spherical support and also related corollaries are stated. March 2004UĞUR ŞENGÜL