The mutual information is analyzed as a function of the input distribution using an identity due to Tops{\o}e for channels with (possibly multiple) linear constraints and finite input and output sets. The mutual information is bounded above by a function decreasing quadratically with the distance to the set of all capacity-achieving input distributions for the case when the distance is less than a certain threshold. Explicit expressions for the threshold and the coefficient of the quadratic decrease are derived. A counter-example is provided demonstrating the non-existence of such a quadratic bound in the case of infinitely many linear cost constraints. Implications of these observations for the channel coding problem and applications of the proof technique to related problems are discussed.

翻译：互信息作为输入分布的函数，利用Topsøe恒等式针对具有（可能多个）线性约束及有限输入输出集的信道进行了分析。当距离小于特定阈值时，互信息的上界由随到所有容量可达输入分布集合的距离呈二次衰减的函数给出。推导了该阈值和二次衰减系数的显式表达式。通过反例证明了在无限多个线性代价约束情形下此类二次上界不存在。讨论了这些发现对信道编码问题的意义，以及该证明技术在其他相关问题中的应用。