Isoperimetric Problems in Discrete Spaces

Abstract

This paper is a survey on discrete isoperimetric type problems. We present here as some known facts about their solutions as well some new results and demonstrate a general techniques used in this area. The main attention is paid to the unit cube and cube like structures. Besides some applications of the isoperimetric approach are listed too. 1 Introduction This paper is devoted to the discrete isoperimetric problem. This problem may be considered as an analog of the well known continuous problem and has some similar features. The discrete isoperimetric problem began to be studied a very long ago and a lot about it's solutions is known now. If there is the only solution of the continuous version, the discrete one, considered for the unit cube, has generally more solutions with much more rich structure, which have no direct continuous analogs. It is mainly due to the facts that, at first not for all values of cardinality of a subset (which is defined as the number of cube points in the..