发表机构
Universidad de Murcia; Universität Bremen（穆尔西亚大学; 不来梅大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个一般原理，将凸锥上满足Brunn-Minkowski不等式的正齐次泛函转化为满足线性Brunn-Minkowski不等式的信息泛函，并建立对偶版本，恢复经典结果并给出$L_p$推广，同时证明齐次性必要性。
AI 中文摘要
本文提出了一个一般原理，表明抽象凸锥上满足适当Brunn-Minkowski不等式的每个正齐次泛函，都能通过将该泛函除以其相对Minkowski容度而定义出一个信息泛函，且该信息泛函满足线性Brunn-Minkowski不等式。我们还建立了该原理对满足某种凸性的泛函的对偶版本，从而得到相关信息泛函的逆向线性Brunn-Minkowski不等式，及其在对偶Brunn-Minkowski理论中的几何应用。作为应用，我们通过一种不同的、一阶的方法恢复了Giannopoulos、Hartzoulaki和Paouris（2002）关于两个连续quermassintegrals之比的经典结果（该结果进一步推广到正定对称矩阵的混合判别式情形），并进一步给出了等号情形的刻画。我们的方法还导出了Dembo、Cover和Thomas关于经典信息泛函的Brunn-Minkowski不等式的一个新的$L_p$（$p \geqslant 1$）推广，其中相对Minkowski容度现在用Lutwak的第一$L_p$ quermassintegral表示。最后，我们通过展示一个与标准高斯测度相关的信息泛函的预期结果的反例，证明齐次性在该原理中是必要的。
英文摘要
In this work we present the following general principle: every positive homogeneous functional on an abstract convex cone satisfying a suitable Brunn-Minkowski inequality gives rise to an information functional -defined as the quotient of the functional by its relative surface area- that satisfies a linear Brunn-Minkowski inequality. We also establish the dual counterpart of this principle for functionals satisfying an analogous convexity property, yielding reverse linear Brunn-Minkowski inequalities for the associated information functionals. Furthermore, we show that homogeneity is necessary within this principle by providing a counterexample to the expected result for the information functional associated to the standard Gaussian measure. As applications, we recover, via a different, first-order argument, a classical result by Giannopoulos, Hartzoulaki and Paouris (2002) for the ratio of two consecutive quermassintegrals, together with a characterization of the equality case. The same argument extends to the setting of mixed discriminants of symmetric positive definite matrices. Finally, our approach also yields a new $L_p$ extension ($p \geq 1$) of the Brunn-Minkowski inequality by Dembo, Cover and Thomas for the classical information functional, where the relative surface area is expressed in terms of Lutwak's first $L_p$ quermassintegral.
CommentsImproved presentation and organization of the manuscript. Some extra remarks and corollaries addeed. Main results unchanged. Corrected minor typos/misprints
