Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities
Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities
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熵的次可加性及其与Brascamp-Lieb型不等式的关系
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
D. Cordero
中科院分区：
文献类型：
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作者：
E. Carlen;D. Cordero
我们证明了一般对偶结果，表明 Brascamp-Lieb 型不等式等价于表达熵的次可加性的不等式，并且最佳常数和等式情况完全对应。这开辟了一种通过熵的次可加性证明 Brascamp-Lieb 型不等式的新方法。我们通过证明表达 $${\mathbb {R}^n}$$ 上熵的次可加性性质的一般不等式，并完全确定等式的情况来说明这种方法的实用性。由于上述二元性，我们获得了经典布拉斯坎-利布不等式的一个简单的新证明，以及所有平等情况的完全明确的确定。我们还推论了一般次可加性不等式的其他几个后果，包括行列式哈达玛不等式的推广。最后，我们还证明了有关费舍尔信息的超可加性和薛定谔算子基本特征值的锐卷积型不等式的第二个对偶定理。虽然我们在本文中主要关注 $${\mathbb {R}^n}$$ 中随机变量的情况，但我们也讨论了其他设置的扩展。
We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on $${\mathbb {R}^n}$$, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp–Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard’s inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schrödinger operators. Though we focus mainly on the case of random variables in $${\mathbb {R}^n}$$ in this paper, we discuss extensions to other settings as well.