A reverse Minkowski-type inequality,Proceedings of the American Mathematical Society

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A reverse Minkowski-type inequality
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2020-07-29 , DOI: 10.1090/proc/15133
Károly J. Böröczky , Daniel Hug

Abstract:The famous Minkowski inequality provides a sharp lower bound for the mixed volume

of two convex bodies $K,M\subset \mathbb{R}^n$ in terms of powers of the volumes of the individual bodies

and

. The special case where

is the unit ball yields the isoperimetric inequality. In the plane, Betke and Weil (1991) found a sharp upper bound for the mixed area of

and

in terms of the perimeters of

and

. We extend this result to general dimensions by proving a sharp upper bound for the mixed volume

in terms of the mean width of

and the surface area of

. The equality case is completely characterized. In addition, we establish a stability improvement of this and related geometric inequalities of isoperimetric-type.

中文翻译：

逆Minkowski型不等式

摘要：著名的Minkowski不等式为两个凸体的混合体积提供了一个尖锐的下界，这取决于单个物体和的体积的幂。特殊情况是单位球会产生等长不等式。在飞机上，Betke和韦尔（1991）发现了一个尖锐的上界的混合区和在的周边方面和。我们通过证明混合体积的平均宽度和表面表面积的尖锐上限将结果扩展到一般尺寸

$K，M \ subset \ mathbb {R} ^ n$

。平等案例的特征已完全体现出来。此外，我们建立了等距类型的此和相关几何不等式的稳定性改进。

更新日期：2020-09-02

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