论文标题
球形几何形状 - 关于凸体宽度和厚度的调查
Spherical geometry -- a survey on width and thickness of convex bodies
论文作者
论文摘要
我们介绍了一篇有关$ d $二维球体$ s^d $的凸形几何形状的调查文章。我们专注于基于凸面$ c \ subset s^d $宽度的概念，由$ c $的支撑半球确定。重要的工具是包含$ c $的月球。支撑性半球接管了凸体在欧几里得空间中支撑的半个空间的作用，并杀死了条带的作用。 $ c $的厚度的概念也至关重要，即其最小宽度。特别是，我们描述了恒定宽度的降低球形凸形体和球形体的特性。最后的概念与$ s^d $上的完整物体和恒定直径的身体的概念相吻合。这里的提醒和评论的结果主要涉及球形凸体的宽度，厚度，直径，周长，区域和极端点，恒定宽度的身体和体降低的身体和体。
We present a survey article about the geometry of convex bodies on the $d$-dimensional sphere $S^d$. We concentrate on the results based on the notion of the width of a convex body $C \subset S^d$ determined by a supporting hemisphere of $C$. Important tools are the lunes containing $C$. The supporting hemispheres take over the role of the supporting half-spaces of a convex body in Euclidean space, and lunes the role of strips. Also essential is the notion of thickness of $C$, i.e, its minimum width. In particular, we describe properties of reduced spherical convex bodies and spherical bodies of constant width. The last notion coincides with the notions of complete bodies and bodies of constant diameter on $S^d$. The reminded and commented here results concern mostly the width, thickness, diameter, perimeter, area and extreme points of spherical convex bodies, reduced bodies and bodies of constant width.
