Hadwiger Covering Functionals And Related Problems

In 1957,Hadwiger made a conjecture that every n-dimensional convex body can be covered by 2ⁿtranslates of its interior.Up to now,this conjecture is still open for all n≥3.In 1933,Borsuk made a conjecture that every n-dimensional bounded set can be divided into n+1 subsets of smaller diameters.Up to now,this conjecture is open for 4≤n≤63.These conjectures are two closely related important topics in discrete geometry.In this thesis,we study the functional forms of them and obtain some new results.Chapter 1.We briefly introduce some known concepts and results which are useful in this thesis.And we review the research background and development status of Hadwiger’s conjecture and Borsuk’s conjecture,and state our main results.Chapter 2.We study Hadwiger covering functionals for simplices and cross-polytopes in n-dimensional Euclid space and obtain the first nontrivial asymptotic upper bounds for them.Chapter 3.We study the Hadwiger covering functionals for the 3-dimensional cross-polytope B₁³in detail.In particular,we obtain the exact values ofγ_m（B₁³）for m=4,...,17,and discover some jumping phenomenon.Chapter 4.We study Borsuk partition functional in Banach space,which is closely related to Hadwiger covering functional.We obtain two methods for esti-mating partition functional and the first nontrivial upper bound forβ（l_p³,8）.These results are closely related to the extension of Borsuk’s problem in finite dimensional Banach spaces and to Zong’s computer program for Borsuk’s conjecture.