  We propose a geometric approach for bounding average stopping times for
stopped random walks in discrete and continuous time. We consider stopping
times in the hyperspace of time indexes and stochastic processes. Our
techniques relies on exploring geometric properties of continuity or stopping
regions. Especially, we make use of the concepts of convex sets and supporting
hyperplane. Explicit formulae and efficiently computable bounds are obtained
for average stopping times. Our techniques can be applied to bound average
stopping times involving random vectors, nonlinear stopping boundary, and
constraints of time indexes. Moreover, we establish a stochastic characteristic
of convex sets and generalize Jensen's inequality, Wald's equations and
Lorden's inequality, which are useful for investigating average stopping times.
