On shortest paths in free spaces including obstacles with fuzzy boundaries

In this paper we introduce definitions of norms and gauges in linear spaces in order to find shortest paths in free spaces with obstacles. Secondly, Euler-Lagrange equations in the calculus of variation give the optimal solutions for the problems of shortest paths. Thirdly we consider a linear structure in a sets of fuzzy numbers and also introduce norms in fuzzy linear spaces. Finally we discuss shortest paths in free spaces including obstacles, with fuzzy boundaries. It is useful in finding shortest paths in realistic environment with natural damage, for example, earthquakes etc.