加权 Berwald 不等式
Weighted Berwald's Inequality
AI总结：
本文将经典 Berwald 不等式推广到 s-凹测度等具有凹性条件的测度上，给出等式成立条件，并应用于凸体与半空间交集的测度界及径向均值体和投影体的推广。
摘要：
Berwald 不等式是对于 ℝⁿ 中凸体上非负凹函数的 p 次平均值（p ∈ (-1,∞)）的一个类似逆 Hölder 的不等式。我们证明了 Berwald 不等式对于关于具有某些凹性条件的测度的函数平均值成立，例如 s-凹测度，s ∈ ℝ。我们还得到了等式成立条件；特别地，这为经典 Berwald 不等式的等式成立条件提供了一个新的证明。作为应用，我们推广了关于凸体与半空间交集测度的若干经典界，并推广了径向均值体和凸体投影体的概念。
英文摘要：
The inequality of Berwald is a reverse-Hölder like inequality for the $p$th average, $p\in (-1,\infty),$ of a non-negative, concave function over a convex body in $\mathbb{R}^n.$ We prove Berwald's inequality for averages of functions with respect to measures that have some concavity conditions, e.g. $s$-concave measures, $s\in \mathbb{R}.$ We also obtain equality conditions; in particular, this provides a new proof for the equality conditions of the classical inequality of Berwald. As applications, we generalize a number of classical bounds for the measure of the intersection of a convex body with a half-space and also the concept of radial means bodies and the projection body of a convex body.
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