Some Geometrical Results in Channel CapacityNachrichtentechnische Zeit, vol. 10. 1957.

A memoryless discrete channel is defined by a set of transition probabilities p i:(j). The rate R of information flow per symbol can be represented as a line segment inside a certain convex body, when the source symbols occur with given probabilities. With the help of this geometric picture one can easily obtain a number of interesting properties of channel capacity and rate of information flow. We show for example that the rate is an upward convex function of the source probabilities, and a strictly convex function of the probabilities of the output symbols. Therefore a local maximum of R is necessarily equal to the absolute maximum C. If ¿ is the rank of the matrix||p i (j)||, then one needs only ¿ (suitably chosen) input symbols to achieve capacity; one has C ¿ log ¿. Other results of a similar nature are developed, including a simple geometric construction of the channel capacity in the case of only two output characters.