关于截面体的凸性研究

Research on the Convexity of Intersection Bodies

著名的Busemann定理表明一个对称凸体的截面体也是凸的.主要研究了Busemann定理对于一般的p-凸体是否也成立,证明了一个对称的p-凸体的截面体对于某些q而言一定是q-凸的;同时,推广了一个Busemann定理,并应用它得到了凸体的截面体的对偶Brunn-Minkowski不等式.

The classical Busemann＇s theorem states that the intersection body of a symmetric convex body is also convex.A version of Busemann＇s theorem for general p-convex bodies is provided.It is shown that the intersection body of a symmetric p-convex body is surely q-convex for some fixed q.Meanwhile,a Busemann＇s theorem is generalized.As an important application,the dual Brunn-Minkowski inequality of the intersection bodies is achieved.