Construction Of The Bodies Of Constant Width

Bodies of constant width are a class of important research object in convex geometry.However,we are only know few examples of the bodies with constant width in Euclidean space.The Blaschke-Lebesgue problem leads to atten-tions from mathematicians,so they study the constructs and geometric properties of the bodies with constant width.In this paper,we mainly discuss the construction of bodies with constant width in Euclidean space.The details are as follows:Firstly,a class of(n-1)-dimensional hypersurfaces“lever surface”?in the Euclidean space and its arm map A are discussed,see chapter 3.Moreover,it is shown that the lever surface is an equivalent characterization of the boundary ofn-dimensional body with constant width,and the parameter representation of the lever surface is established by(?)where?and A satisfy(?)Secondly,based on the equivalent characterization and the parameter represen-tation of the bodies with constant width in R~n,the parameter equation and the area formula of the general planar curve of constant width are given in chapter 4.Mean-while,a class of planar constant width curves are constructed by the trigonometric function.It is also proven that the Reuleaux polygons are a class of constant width curves with the piecewise constant arm function.Then,a general constant width polygons are constructed by solving the feasible region of the linear constraint.Finally,based on the equivalent characterization and the parameter representa-tion of the bodies with constant width in R~n,a sufficient condition for continuous constant width surfaces in R~3is obtained,then parameter representation of the sec-ond order differentiable constant width surfaces in R~3is given.Moreover,a class of constant width surfaces in the R~3is constructed by the trigonometric function.