We show that the uniform distribution minimizes entropy among all one-dimensional symmetric log-concave distributions with fixed variance, as well as various generalizations of this fact to Rényi entropies of orders less than 1 and with moment constraints involving $p$ -th absolute moments with $p\leq 2$ . As consequences, we give new capacity bounds for additive noise channels with symmetric log-concave noises, as well as for timing channels involving positive signal and noise where the noise has a decreasing log-concave density. In particular, we show that the capacity of an additive noise channel with symmetric, log-concave noise under an average power constraint is at most 0.254 bits per channel use greater than the capacity of an additive Gaussian noise channel with the same noise power. Consequences for reverse entropy power inequalities and connections to the slicing problem in convex geometry are also discussed.

中文翻译：

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对称对数凹面分布的锐矩熵不等式和容量界限

我们表明，均匀分布最小化了所有具有固定方差的一维对称对数凹面分布之间的熵，以及这一事实对小于 1 阶的 Rényi 熵和矩约束的各种推广，包括 $p$ -th绝对时刻 $p\leq 2$ . 结果，我们为具有对称对数凹面噪声的加性噪声​​通道以及涉及正信号和噪声的时序通道提供了新的容量界限，其中噪声具有递减的对数凹面密度。特别是，我们表明，在平均功率约束下，具有对称对数凹面噪声的加性噪声​​信道的容量比具有相同噪声功率的加性高斯噪声信道的容量大每信道使用最多 0.254 位。还讨论了逆熵功率不等式的后果以及与凸几何中切片问题的联系。