Nonconvexity of the capacity region of the multiple-access arbitrarily varying channel subject to constraints

The multiaccess arbitrarily varying channel (MAVC) subject to state and input constraints models a channel that is subject to jamming or unknown, time-varying uncertainties. The random-code capacity region has been characterized under two different assumptions. Under the assumption that the random encoders are statistically independent, the capacity region is completely characterized. In addition, it is shown by means of a simple erasure channel that this capacity region, 𝒞, is not always convex. To our knowledge, this is the first example of a synchronous multiple-access channel with a nonconvex capacity region. When the random encoders are permitted to be statistically dependent, the capacity region is denoted by 𝒞¯. Clearly 𝒞⊆𝒞¯. It is known that 𝒞¯ is always convex. By considering another erasure channel, it is shown that even when 𝒞 is convex, it is possible for 𝒞 to be a proper subset of 𝒞¯. In fact, the set-theoretic difference, 𝒞¯/𝒞, has a nonempty interior. Hence, permitting the codebooks to be dependent can increase the capacity region