$\alpha$-concave functions and a functional extension of mixed volumes

Q3 Mathematics
V. Milman, Liran Rotem
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引用次数: 17

Abstract

Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation $\oplus$ on the class of quasi-concave functions, such that every class of $\alpha$-concave functions is closed under $\oplus$. We then define the mixed integrals, which are the polarization of the integral with respect to $\oplus$. We proceed to discuss the extension of various classic inequalities to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to $\alpha$-concave functions, we state a generalization of the Alexandrov inequalities in their more familiar form.
$\alpha$-凹函数和混合体的功能扩展
混合体积是体积相对于闵可夫斯基加法的极化,是凸性的基本对象。在这篇笔记中,我们宣布混合积分的构造,它是混合体积的泛函类似物。我们在拟凹函数类上建立了一个自然加法运算$\oplus$,使得每一类$\alpha$ -凹函数都在$\oplus$下闭合。然后我们定义混合积分,它是关于$\oplus$的积分的极化。我们继续讨论各种经典不等式在函数设置中的推广。对于一般的拟凹函数,这是通过用重排不等式的语言重述这些结果来完成的。限制我们自己$\alpha$ -凹函数,我们以更熟悉的形式陈述Alexandrov不等式的推广。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
0.90
自引率
0.00%
发文量
0
审稿时长
>12 weeks
期刊介绍: Electronic Research Archive (ERA), formerly known as Electronic Research Announcements in Mathematical Sciences, rapidly publishes original and expository full-length articles of significant advances in all branches of mathematics. All articles should be designed to communicate their contents to a broad mathematical audience and must meet high standards for mathematical content and clarity. After review and acceptance, articles enter production for immediate publication. ERA is the continuation of Electronic Research Announcements of the AMS published by the American Mathematical Society, 1995—2007
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