  Memoryless channels with synchronization errors as defined by a stochastic
channel matrix allowing for symbol insertions and deletions in addition to
random errors are considered. Such channels are information stable, hence their
Shannon capacity exists. However, computation of the channel capacity is
formidable, and only some upper and lower bounds on the capacity (for some
special cases) exist. In this short paper, using a simple methodology, we prove
that the channel capacity is a convex function of the stochastic channel
matrix. Since the more widely studied model of an independent identically
distributed (i.i.d.) deletion channel is a particular case, as an immediate
corollary to this result we also argue that the i.i.d. deletion channel
capacity is a convex function of the deletion probability. We further use this
result to improve the existing capacity upper bounds on the deletion channel by
a proper "convexification" argument. In particular, we prove that the capacity
of the deletion channel, as the deletion probability d --> 1, is upper bounded
by $0.4143(1-d)$ (which was also observed by a different (weaker) recent
result).
