We consider the problem of finding linear decoders to achieve the capacity of compound channels. Here, a linear decoders is one based on a decoding metric that is additive over time. Mathematically, linear decoding defined this way is equivalent as a linear decision over the space of empirical distributions. A natural geometric approach is applied to this problem. It is found that in general the compound capacity is NOT achieved by GLRT, but rather by a slightly modified version, which is called the generalized MAP test. We develop intuitive interpretation of this result by discussing the very noisy case of the problem, in which case the problem is reduced to Euclidean geometry.