Geometric Representation of Classes of Concave Functions and Duality

Grigory Ivanov, Elisabeth M. Werner
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Abstract

Using a natural representation of a 1/s-concave function on \({\mathbb {R}}^d\) as a convex set in \({\mathbb {R}}^{d+1},\) we derive a simple formula for the integral of its s-polar. This leads to convexity properties of the integral of the s-polar function with respect to the center of polarity. In particular, we prove that the reciprocal of the integral of the polar function of a log-concave function is log-concave as a function of the center of polarity. Also, we define the Santaló regions for s-concave and log-concave functions and generalize the Santaló inequality for them in the case the origin is not the Santaló point.

凹函数类的几何表示和对偶性
利用在 \({\mathbb {R}}^{d+1},\) 上的 1/s-concave 函数作为凸集的自然表示,我们得出了其 s-polar 积分的简单公式。这引出了 s 极函数关于极性中心的积分的凸性性质。特别是,我们证明了对数凹函数的极值函数积分的倒数作为极性中心的函数是对数凹的。此外,我们还定义了 s-concave 和 log-concave 函数的 Santaló 区域,并在原点不是 Santaló 点的情况下推广了它们的 Santaló 不等式。
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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