Convex Hull And Convex Body That Contains The Measure

Convex bodies are the research objects in the paper. The paper is concerned with two aspects:A necessary and sufficient condition for convex hull of a set in E2 being closed; Convex body and its contain measure. This essay is organized as follows:① if A is an open set,then convA is open . Thus whether or not the convex hull of aclosed set is also closed? The answer is clearly negative .For example, in E~2 , if A = {(x,y)|x~2y~2 =1,y >0},then A is a closed set,but convA = {(x,y)|y>0} is not closed!!Which set satisfies its convex hull is closed ?What is the character of a set that its convex hull is closed ?That is to say what is the necessary and sufficient condition for convex hull of a set in E2 being closed . This topic justly gives out a necessary and sufficient condition for convex hull of a set in E2 which is closed through the research of convex hull of a set in E2 and two new concepts : critical supporting line and left critical interior point and. right critical interior point② Convex body and its contain measureTwo new concepts, the generalized support function and restricted chord function,both referring to the convex set, have been introduced in Reference. General formulate to yield the kinematic measure of a segment of fixed length in a convex set were established based on these concents and a discussion was given with respect to rectangular domain. The Reference deals with some otherwise shaped figures: the parallelogram, triangle and regular hexagon. As for these figures, the authors succeed in giving their generalized support functions and restricted chord functions, and deriving the individual expressions for the kinematic measure mentioned above. A way called part intersection with strong lives and the generalized support function and restricted chord function has been used in this topic. We use the simple maths and the ralated theory of measure to derive the individual expressions for the kinematic measure of the other convex polygon (ellipse).