Holmes-Thompson边界面积的Santaló点

On Explicit Holmes-Thompson Area Formula in Integral Geometry

In this article, we give an exposition on the Holmes-Thompson theory developed by Alvarez. The space of geodesics in Minkowski space has a symplectic structure which is induced by the projection from the sphere-bundle. we show that it can be also obtained from the symplectic structure on the tangent bundle of the Riemannian manifold, the tangent bundle of the Minkowski unit sphere. We give detailed descriptions and expositions on Holmes-Thompson volumes in Minkowski space by the symplectic structure and the Crofton measures for them. For the Minkowski plane, a normed two dimensional space, we …

霍姆斯-汤普森面积与内接多边形及凸投影结构

Holmes-Thompson area of inscribed polygons and convex projective structures

本文研究了正向旗在三维空间中定义的凸多边形，并探讨了霍姆斯-汤普森面积与希尔伯特度量之间的关系，以及有限面积凸实投影结构的参数化问题。

凸体的幅度与Holmes-Thompson内蕴体积

Magnitude and Holmes-Thompson intrinsic volumes of convex bodies

本文证明了超度量赋范空间中凸体幅度可由Holmes-Thompson内蕴体积控制的上界，并借此给出Mahler猜想zonoid情形与Sudakov最小值不等式的新证明。

Filling minimality of Finslerian 2-discs

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this result for Finsler metrics with area defined as the two-dimensional Holmes-Thompson volume. This implies a generalization of Pu's isosystolic inequality to Finsler metrics, both for Holmes-Thompson and Busemann definitions of Finsler area.

On filling minimality of simple Finsler manifolds

This paper states a formula for the difference of the Holmes-Thompson volumes of two simple Finsler manifolds of arbitrary dimension, in terms of the boundary distances and their derivatives. An application is a preconditioned filling minimality result.

具有极小测地线的Finsler圆盘的最小面积

Minimal area of Finsler disks with minimizing geodesics

该研究证明内部测地线均为长度极小的Finsler圆盘的Holmes--Thompson面积下界，构造紧例并发现非旋转对称度量可达等号，通过随机测地线离散化度量证明定理，还得到相关积分几何公式在特定Finsler流形上成立的辅助结果。