It is well known that there is a strong connection between entropy inequalities and submodularity, since the entropy of a collection of random variables is a submodular function. Unifying frameworks for information inequalities arising from submodularity were developed by Madiman and Tetali (2010) and Sason (2022). Madiman and Tetali (2010) established strong and weak fractional inequalities that subsume classical results such as Han's inequality and Shearer's lemma. Sason (2022) introduced a convex-functional framework for generalizing Han's inequality, and derived unified inequalities for submodular and supermodular functions. In this work, we build on these frameworks and make three contributions. First, we establish convex-functional generalizations of the strong and weak Madiman and Tetali inequalities for submodular functions. Second, using a special case of the strong Madiman-Tetali inequality, we derive a new Loomis-Whitney-type projection inequality for finite point sets in $\mathbb{R}^d$, which improves upon the classical Loomis-Whitney bound by incorporating slice-level structural information. Finally, we study an extremal graph theory problem that recovers and extends the previously known results of Sason (2022) and Boucheron et al., employing Shearer's lemma in contrast to the use of Han's inequality in those works.

翻译：众所周知，熵不等式与子模性之间存在紧密联系，因为一组随机变量的熵是子模函数。Madiman 与 Tetali (2010) 以及 Sason (2022) 分别建立了源于子模性的信息不等式的统一框架。Madiman 与 Tetali (2010) 建立了强与弱分数阶不等式，这些不等式涵盖了 Han 不等式和 Shearer 引理等经典结果。Sason (2022) 引入了推广 Han 不等式的凸函数框架，并推导了子模函数与超模函数的统一不等式。本文基于这些框架，做出了三项贡献。首先，我们针对子模函数建立了强与弱 Madiman-Tetali 不等式的凸函数推广。其次，利用强 Madiman-Tetali 不等式的一个特例，我们推导了 $\mathbb{R}^d$ 中有限点集的一个新的 Loomis-Whitney 型投影不等式，该不等式通过纳入切片层面的结构信息，改进了经典的 Loomis-Whitney 界。最后，我们研究了一个极值图论问题，该问题恢复并扩展了 Sason (2022) 和 Boucheron 等人先前已知的结果，其中我们运用了 Shearer 引理，而上述工作中使用的是 Han 不等式。