Volumes in normed vector spaces and Finsler manifolds

赋范空间和芬斯勒流形中体积的凸性性质涉及到凸性、几何不等式和几何分析等基本问题。在这个项目中，将研究不同体积概念的凸性，如Busemann体积，Holmes-Thompson体积，以及最近引入的基于质心体的体积。此外，还将研究Finsler流形、极小子流形和极值子流形以及诸如推测的Petty投影不等式等几何不等式上封闭测大地长度的凸性的结果。

Convexity properties of volumes in normed spaces and Finsler manifolds are related to fundamental problems in convexity, geometric inequalities and geometric analysis. In this project, convexity of different notions of volumes, like Busemann volume, Holmes-Thompson volume, and the recently introduced volume based on centroid bodies will be studied. Moreover, consequences of convexity for lengths of closed geodesics on Finsler manifolds, minimal and extremal submanifolds and geometric inequalities like the conjectured Petty's projection inequality will be investigated.