关于凸体覆盖的Hadwiger猜想

针对Hadwiger猜想这一凸和离散几何中持续了近60年的著名难题,回顾了这一猜想的提出,它的若干等价形式,以及前人在尝试解决这一猜想的过程中取得的阶段性进展.介绍了该猜想相关的若干研究问题及相应结果.结果表明:一方面,Hadwiger猜想仍是一个远未解决的公开难题;另一方面,围绕着Hadwiger猜想及其相关问题还有大量亟待完成的基础性研究工作。     

关于斜范数的Dvoretzky—Rogers定理

在所谓的斜范空间上建立了一个Ｄｖｏｒｅｔｚｋｙ－Ｒｏｇｅｒｓ型定理，为有限维空间中一般凸体几何性质的研究提供了一个理论工具。     

Cavitation bubbles collapse characteristics behind a convex body

Cavitation bubbles behind a convex body were experimentally studied by a high speed camera and a hydrophone synchronously. The experiments were conducted in a circulating water tunnel with five various contraction ratios： β = 0.497, β= 0.6, β= 0.697, β= 0.751, and β= 0.799. The distributions of the cavitation bubble collapse positions behind the five different convex bodies were obtained by combining the images taken by the high speed camera. According to the collapse positions, it was found that no cavitation bubble was collapsed in the region near the wall until the ratio of the water head loss over the convex body height was larger than 20, which can be used to predict if the cavitation damage would occur in the tunnel with orifice energy dissipaters.     

An alternate formulation of mean value for random geometric figures*

In contrast to the traditional technique of averaging functionals of a random geometric figure, it is possible to consider an averaging process which treats the figure as a whole. The result in probabilistic terminology is an expectation, which is itself a figure. Theoretical foundations and various instances are presented in an informal survey.