The Geometrical Measure Of Embedding A Simplicial Complex Into Euclidean Space

Every k-dimensional simplicial complex can be linearly embedded into a Euclidean space of dimension2k+1.The paper concerns the geometrical complexity of embedding a simplicial complex into Euclidean space since there are varieties of methods to embed.The notion of point-surface thickness of a linear embedding is introduced which allows us to obtain other geometric information of the linear embedding. In particular,a sharp inequality between the point-surface thickness of linear embedding and the thickness in the sense of Gromov and Guth is established.In addition, we give a lower bound of the thickness for the linear embedding defined by the moment curve.This thesis is made up of five parts,and the arrangement is as follows.In Part l,we briefly introduce the background,present situation and research meaning for embedding a simplicial complex into Euclidean space. Our main conclusions are also included.In part2,we introduce some basic knowledge associated with our paper in order to prove part3and part5.In Part3,the proof of Theorem1.1.In Part4,we introduce some basic concepts of n-dimensional Eu-clidean space.Two examples are gave to confirm that the inequality of Theorem1.1(1) is sharp.In Part5,the proof of Theorem1.2.