Simple Type of Support Functions of Fuzzy Sets and Its Applications

The support functions of crisp sets have been widely used in the topic of nonlinear analysis. This observation inspires to introduce the support functions of fuzzy sets. Two different support functions of fuzzy sets were introduced in the literature by using different domains of functions. This paper will focus on the simple type of support functions that are defined on a simple domain. Three applications using the simple type of support functions are studied in this paper. The first one is to generate fuzzy sets from the real-valued functions that satisfy some regular conditions such that the simple type of support function of a generated fuzzy set is identical with the pre-determined real-valued function. The second one is to study the embedding theorems of some interesting families of fuzzy sets. In this case, the continuities of simple types of support functions should be studied under some suitable topologies. In order to make the embedded Banach spaces as small as possible, the concept of weak* topology should be involved in the study. The third one is to solve the fuzzy optimization problems using the embedding theorems.