Mathematical Sciences: Isoperimetric Inequalities

Erwin Lutwak将继续他对经典几何不等式的研究。注意力将集中在建立一个与布鲁恩-明科夫斯基理论对偶的理论上。当应用于凸体投影问题时，后一种理论被证明是非常有效的。Lutwak用对偶理论的目标是开发适用于相交问题的技术。近年来，Brunn-Minkowski混合体积理论已被证明具有适用于星形集的对偶理论。这主要是由于使用径向函数而不是集合的支持函数。许多关于投影和投影体的经典结果在相交和相交体方面有其对偶。这种对偶性是研究交叉口明显异常行为的有用工具。

Erwin Lutwak will continue his investigations of the classical geometric inequalities. Attention will be focussed on establishing a theory dual to the Brunn-Minkowski theory. This latter theory has proved very powerful when applied to problems regarding projections of convex bodies. Lutwak's objective with the dual theory is to develop techniques applicable intersection problems. The Brunn-Minkowski theory of mixed volumes has in recent years been shown to have a dual theory applicable to starshaped sets. This arises essentially from use of the radial function rather than the support function of the set. Many of the classical results for projections and projection bodies have their duals in terms of intersections and intersection bodies. This duality is a useful tool in the study of the apparently anomalous behaviour of intersections.