凸性条件下随机向量每坐标熵受到严格约束
The entropy per coordinate of a random vector is highly constrained under convexity conditions
AI总结：
本文证明任意对数凹随机向量的每坐标熵与同最大密度正态随机变量之差至多1/2，并推广到凸或双曲概率测度，给出超平面猜想的信息论表述及相关熵界应用。
摘要：
本文证明，对于任意维度、且在众数处具有给定密度的对数凹随机向量，其每坐标熵的取值范围仅为1。凸体上的均匀分布位于该范围的下端，坐标独立同分布且服从指数分布的分布位于上端，而正态分布恰好位于中间。因此，就每坐标熵所度量的随机性而言，任意维度的任意对数凹随机向量所含随机性与具有相同最大密度值的正态随机变量的随机性之差至多为1/2。作为应用，我们得到了凸几何中著名的超平面猜想的信息论表述、某些无限可分分布的熵界，以及卷积下众数处密度行为的定量估计。更一般地，可以考虑欧氏空间上所谓的凸或双曲概率测度；我们为这类测度给出了新的每坐标熵约束，这些约束推广了我们在对数凹假设下的结果，揭示了多元Pareto型分布的极值作用，并给出了一些应用。
英文摘要：
The entropy per coordinate in a log-concave random vector of any dimension with given density at the mode is shown to have a range of just 1. Uniform distributions on convex bodies are at the lower end of this range, the distribution with i.i.d. exponentially distributed coordinates is at the upper end, and the normal is exactly in the middle. Thus in terms of the amount of randomness as measured by entropy per coordinate, any log-concave random vector of any dimension contains randomness that differs from that in the normal random variable with the same maximal density value by at most 1/2. As applications, we obtain an information-theoretic formulation of the famous hyperplane conjecture in convex geometry, entropy bounds for certain infinitely divisible distributions, and quantitative estimates for the behavior of the density at the mode on convolution. More generally, one may consider so-called convex or hyperbolic probability measures on Euclidean spaces; we give new constraints on entropy per coordinate for this class of measures, which generalize our results under the log-concavity assumption, expose the extremal role of multivariate Pareto-type distributions, and give some applications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
