We consider the binary independent identically distributed (i.i.d.) deletion channel where each bit is 
either received correctly or deleted with probability d independently of any other. Neither the transmitter 
nor the receiver has knowledge of the positions of the deletions. A classic result due to Dobrushin assures 
that this channel is information stable, and its Shannon capacity exists. In this presentation, we prove 
using a simple argument that its capacity is a convex function of the deletion probability d, and discuss 
a few implications of this result. In particular, we extend the best known upper bounds on the channel 
capacity by a proper “convexification” argument, and obtain tighter bounds. We also prove that as d approaches
unity, the capacity is upper bounded by 0.42(1-d) (which was also observed by a different (weaker) recent 
result). We further talk about a few extensions, including the convexity of the channel capacity for more 
general channel models (e.g., i.i.d. deletion/substitution channel) in the channel parameters involved.