Some Characteristic Properties Of Convex Set And On Buffon's Problem For The Complicated Lattice

Convex bodies are the research objects in the paper. The paper is concerned with two aspects: a new concept of convex set and its some characteristic properties; on Buffon's problem for the complicated lattice.(1) A new concept of convex set and its some characteristic properties This paper give a new definition, the support direction of a boundary point of convex set. Then use it to prove a characteric properties of convex set and some other characteric properties.(2) On Buffon's problem for the complicated lattice Kinematic formulas in integral geometry are integral formulas that represent integrals of geometric functions on the intersection of fixed and moving domains. These formulas can be viewed as integral formulas for various intersection measures. They are useful for solving problems in geometric probabilities. Some problems in geometric probability require more tools other than intersection measures. For instance, solutions to the Buffon needle problem of lattices need to compute the containment measure.In this paper, by using the kinematic measure for a segment of fixed length within a special convex domain, we obtain the Buffon probability formula of grid whose basic regional are two different convex domains, and its outreach to various convex domain cases.