Quasi-support hyperplanes in asymmetric normed spaces

The hyperplane theory plays a very important role in the research of optimization, it can help us better understand and solve various optimization problems. Therefore, the development of hyperplane theory has always been concerned by the scholars. This paper mainly studies the quasi-support hyperplane, which is the generalization of a support hyperplane, on the asymmetric normed space. First, the properties of the distance from a point to a non-empty set are studied more comprehensively, the isometric property of the distance from a point to a half space and to the corresponding hyperplane is proved. Second, the problem of supremum (infimum, resp.) of a linear function on a set is reduced to the problem of upper quasi-support (lower quasi-supports, resp.) hyperplane of the set. Additionally, some equivalent conditions for the quasi-support hyperplane are given by using the theory of dual space of asymmetric normed spaces. The obtained results enrich the functional analysis in asymmetric normed spaces and have great significance to applications in optimization theory, approximation problems, complexity analysis and the other fields.