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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  ε -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.

Subdifferential properties of a perturbed minimal time function in normed spaces