Subdifferential properties of a perturbed minimal time function in normed spaces

In a normed space, we study the perturbed minimal time function determined by a bounded closed convex set and a proper lower semicontinuous function. It covers the perturbed distance function as a special case. In particular, we show that the epsilon-subdifferential and the limiting subdifferential of the perturbed minimal time function are representable by virtue of corresponding subdifferential of the associated function and the support function of the constraint set.