Properties Of The Maximal Time Function In Normed Spaces

In normed vector spaces,the properties of the maximal time function are studied.The maximal time function which plays an important role in geometry covers the farthest distance function as a special case and has a close relationship with the smallest enclosing ball problem.The maximal time function is determined by a target set and a nonempty control set.Firstly,when the target set is fixed,it is proved that the maximal time function is convex and lower semicontinuous,and the subdifferential of the maximal time function can be expressed as the normal cone of the sublevel set and the level set of the support function of the control set.Secondly,when the target set is not fixed,the maximal time function with moving target sets is no longer convex.Therefore,we establish the upper and lower estimates for the Fr(?)chet subdifferential of the maximal time function with moving target sets being representable by virtue of the appropriate normal cone and the support function of the control set.