Volumes in normed vector spaces and Finsler manifolds

赋范空间和Finsler流形中体的凸性性质涉及到凸性、几何不等式和几何分析中的基本问题。在这个项目中，凸性的不同概念的体积，如Busemann体积，霍姆斯-汤普森体积，和最近推出的基于质心体的体积将进行研究。此外，Finsler流形，极小和极值子流形和几何不等式，如固定的Petty投影不等式的闭测地线的长度的凸性的后果将被调查。

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.