Homeomorphism conditions for coherently oriented piecewise affine mappings

• 成果类型: Article
• 署名作者: Scholtes, S
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.21.4.955
• 发表日期: 1996
• 页码: 955-978
• 关键词: normal maps

摘要

This article is mainly concerned with the homeomorphism problem for piecewise affine mappings (PA-maps), i.e., mappings which coincide with an affine mapping on each polyhedron of some finite polyhedral subdivision of R(n). In the first part, we prove that a PA-map can be defined without referring to a subdivision of R(n) as a continuous mapping which coincides at every point x is an element of R(n) with al least one function from a finite collection of affine functions. The second part studies the recession function of a PA-map. It is shown that the recession function is piecewise linear and that a coherently oriented PA-map is a homeomorphism if and only if its recession function is a homeomorphism. In the last part we prove that a coherently oriented PA-map is a homeomorphism if it admits a corresponding polyhedral subdivision of R(n) such that for some number k is an element of {2,..., n} every face of codimension k is contained in at most 2k polyhedra, provided the subdivision contains at least one face of codimension k.